Sequence and Series

Find the sum of n terms of the A.P., whose nth term is $\displaystyle 5n + 1$

The sum of first three terms of a G.P. is $\displaystyle \frac{21}{2}$ and their product is 27. Which of the following is not a term of the G.P., if the numbers are positive?

Insert 4 numbers between 2 and 22 such that the resulting sequence is an Arithmetic Progression (A.P.).

Find the sum of series $\displaystyle 1 + \frac{1}{2} + \frac{1}{4} + \dots$ upto 6 terms.

If the sum of '$\displaystyle n$' terms of an AP (Arithmetic Progression) is $\displaystyle 2n^2$, the fifth term is ____.

The sum of first $\displaystyle n$ terms an AP is $\displaystyle 3n^2+5n$. The series is:

If $\displaystyle 2+6+10+14+.........+x=882$ then the value of $\displaystyle x$ is

If the sum of five terms of AP is $\displaystyle 75$. Find the third term of the series.

The $\displaystyle 20^{th}$ term of arithmetic progression whose $\displaystyle 6^{th}$ term is $\displaystyle 38$ and $\displaystyle 10^{th}$ term is $\displaystyle 66$ is:

Divide $\displaystyle 69$ into $\displaystyle 3$ parts which are in A.P. and are such that the product of first two parts is $\displaystyle 483$.

The number of terms of the series: $\displaystyle 5+7+9+...$ must be taken so that the sum may be $\displaystyle 480$.

If the sum of $\displaystyle n$ terms of an AP is $\displaystyle (3n^2-n)$ and its common difference is $\displaystyle 6$, then its $\displaystyle 1^{st}$ term is:

Insert two arithmetic means between $\displaystyle 68$ and $\displaystyle 260$.

If the $\displaystyle p^{th}$ term of an A.P. is '$\displaystyle q$' and the $\displaystyle q^{th}$ term is '$\displaystyle p$', then its $\displaystyle n^{th}$ term is

The sum of the series $\displaystyle -8,-6,-4,.........n$ terms is $\displaystyle 52$. The number of terms $\displaystyle n$ is

The value of $\displaystyle K$, for which the terms $\displaystyle 7K+3, 4K-5, 2K+10$ are in A.P., is

If $\displaystyle p^{th}$ term of an AP is $\displaystyle q$ and its $\displaystyle q^{th}$ term is $\displaystyle p$, then what will be the value of $\displaystyle (p+q)^{th}$ term?

How many numbers between $\displaystyle 74$ and $\displaystyle 25,556$ are divisible by $\displaystyle 5$?

If $\displaystyle 9^{th}$ and $\displaystyle 19^{th}$ term of an AP are $\displaystyle 35$ and $\displaystyle 75$, respectively, then its $\displaystyle 20^{th}$ term is:

Find the $\displaystyle 17^{th}$ term of an AP series if $\displaystyle 15^{th}$ and $\displaystyle 21^{st}$ terms are $\displaystyle 30.5$ and $\displaystyle 39.5$ respectively.

If $\displaystyle n^{th}$ term of an AP series is $\displaystyle 7n-2$, then sum of '$\displaystyle n$' terms is:

Find the value of '$\displaystyle x$' for the following data $\displaystyle 1+7+13+19.........+x=225$

If fourth term of A.P. series is zero, then what is the ratio of twenty-fifth term of eleventh term?

If $\displaystyle 8^{th}$ term of an AP is $\displaystyle 15$, the Sum of the $\displaystyle 15$ its term is

For what value of $\displaystyle x$; the sequence $\displaystyle x+1, 3x, 4x+2$ are in AP?

The $\displaystyle n^{th}$ element of the series $\displaystyle 3,5,7,...$ is

If $\displaystyle 1+3+5+.........+n$ terms $\displaystyle = \frac{n^2}{2}$ If $\displaystyle 2+4+6+.........+50$ terms $\displaystyle = 51$ then the value of '$\displaystyle n$' is

If $\displaystyle 6^{th}$ and $\displaystyle 13^{th}$ term of an A.P are $\displaystyle 15$ and $\displaystyle 36$ respectively the A.P is

The value of $\displaystyle K$, for which the terms $\displaystyle 7K+3$, $\displaystyle 4K-5$, $\displaystyle 2K+10$ are in A.P., is

The first term of an A.P. is $\displaystyle 100$ and the sum of whose first $\displaystyle 6$ terms is $\displaystyle 5$ times the sum of the next $\displaystyle 6$ terms, then the c.d. is:

The sum of $\displaystyle n$ terms of an A.P. is $\displaystyle 3n^2+n$; then its $\displaystyle p^{th}$ term is

If three AM's between $\displaystyle 3$ and $\displaystyle 11$, they are

The first and the last term of an AP are $\displaystyle -4$ and $\displaystyle 146$. The sum of the terms is $\displaystyle 7171$. The number of terms is

The sum of the first $\displaystyle 3$ terms in an AP is $\displaystyle 18$ and that of the last $\displaystyle 3$ is $\displaystyle 28$. If the AP has $\displaystyle 13$ terms, find sum of the middle three terms?

The ratio of sum of first $\displaystyle n$ natural numbers to sum of cubes of first $\displaystyle n$ natural numbers is

Sum of progression $\displaystyle (a+b)$, $\displaystyle (a-b)$, $\displaystyle (a-3b)...n$ term is

Find the sum of first twenty-five terms of A.P. series whose $\displaystyle n^{th}$ term is $\displaystyle \left(n+\frac{2}{5}\right)$

Insert $\displaystyle 4$ A.M.'s between $\displaystyle 3$ and $\displaystyle 18$:

The first and fifth term of an A.P. of $\displaystyle 40$ terms are $\displaystyle -29$ and $\displaystyle -15$ respectively. Find the sum of all positive terms of this A.P

If the common difference of an AP equals to the first term, then the ratio of its $\displaystyle m^{th}$ term and $\displaystyle n^{th}$ term is:

Find the value of $\displaystyle 1+2+3+...+105$

The $\displaystyle n^{th}$ term of the series $\displaystyle 9,7,5,...$ and $\displaystyle 15,12,9,...$ are same. Find the $\displaystyle n^{th}$ term?

If the ratio of sum of $\displaystyle n$ terms of two APs is $\displaystyle (n-1):(n+1)$, then the ratio of their $\displaystyle n^{th}$ terms is

In an arithmetic progression, the seventh terms is $\displaystyle x$, and $\displaystyle (x+7)^{th}$ term is zero. Then $\displaystyle x^{th}$ term is

If the second and eight terms of an arithmetic progression (AP) are equal to constant $\displaystyle a$, then the sum of first $\displaystyle n$ terms of this AP is equal to

The $\displaystyle 3^{rd}$ term of arithmetic progression is $\displaystyle 7$ and Seventh term is $\displaystyle 2$ more than thrice of third term. The common difference is

Which term of the AP $\displaystyle 64, 60, 56, 52...$ is $\displaystyle 0$

If the sum of $\displaystyle n$ terms of an AP is $\displaystyle 2n^2$, then $\displaystyle 5^{th}$ term is

Sum up to infinity of series. $\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots$

Sum the series $\displaystyle \frac{1}{5} + \frac{1}{5^2} + \frac{1}{5^3} + \dots + \frac{1}{5^n}$

Find no. of terms of the series $\displaystyle 25, 5, 1, \dots, \frac{1}{3125}$

Three numbers in G.P. with their sum 130 and their product 27,000 are:

In a geometric progression, that $\displaystyle 3^{rd}$ and $\displaystyle 6^{th}$ terms are respectively 1 and $\displaystyle -1/8$. The term $\displaystyle (a)$ and common ratio are respectively.

If the sum and product of three numbers in G.P. are 7 and 8 respectively, then $\displaystyle 4^{th}$ term of the series is

The $\displaystyle 3^{rd}$ term of a G.P. is $\displaystyle 2/3$ and the $\displaystyle 6^{th}$ term is $\displaystyle 2/81$ then the $\displaystyle 1^{st}$ term is

The sum of series $\displaystyle 7 + 14 + 21 + \dots$ to $\displaystyle 17^{th}$ term is:

The largest value of $\displaystyle n$ for which $\displaystyle 1 + \frac{1}{2} + \frac{1}{2^2} + \dots + \frac{1}{2^{n-1}} < 0.998$ is

The sum of first 8 terms of a G.P is five times the sum of the first 4 terms. Find the common ratio?

In a GP $\displaystyle 5^{th}$ term is 27 and $\displaystyle 8^{th}$ term is 729. Find its $\displaystyle 11^{th}$ term?

If $\displaystyle 4^{th}, 7^{th}$ and $\displaystyle 10^{th}$ terms of a Geometric Progression are $\displaystyle p, q$ and $\displaystyle r$, respectively, then:

Given an infinite geometric series with first term '$\displaystyle a$' and common ratio '$\displaystyle r$'. If its sum is 4 and the second term is $\displaystyle \frac{3}{4}$, then one of correct option is

If for an infinite geometric progression, first term is '$\displaystyle a$', common ratio is '$\displaystyle r$', the sum is 8 and the second term is $\displaystyle \frac{7}{8}$, then

For what values of $\displaystyle X$, the number $\displaystyle \frac{-2}{7}, X, \frac{-7}{2}$ are in G.P.?

If the sum of $\displaystyle n$ terms of an AP is $\displaystyle 3n^2-n$ and its common difference is $\displaystyle 6$, then its $\displaystyle 1^{st}$ term is

Sum lying from $\displaystyle 100$ to $\displaystyle 300$ which is divisible by $\displaystyle 4$ and $\displaystyle 5$ is

Sum of $\displaystyle n$ terms of two AP's are in the ratio $\displaystyle (3n+5):(5n+3)$ then ratio of their $\displaystyle 10^{th}$ term

If the $\displaystyle p^{th}$ term of an A.P. is 'q' and the $\displaystyle q^{th}$ term is 'p', then its $\displaystyle r^{th}$ term is:

In AP $\displaystyle T_p=q$ and $\displaystyle T_q=p$ then $\displaystyle T_{p+q}=$

If the sum of $\displaystyle n$ terms of an A.P. be $\displaystyle 2n^2+5n$, then its '$\displaystyle n^{th}$' term is:

If $\displaystyle 20$ AMs. are inserted between $\displaystyle 3$ and $\displaystyle 51$ then sum of these $\displaystyle 20$ A.M.s is

The first, second and seventh term of an AP are in G.P. and the common difference is $\displaystyle 2$, the $\displaystyle 2^{nd}$ term of A.P. is:

The $\displaystyle 4^{th}$ term of an A.P. is three times the first and the $\displaystyle 7^{th}$ term exceeds twice the third term by $\displaystyle 1$. Find the first term 'a' and common difference 'd'.

Find the sum of all natural numbers betn $\displaystyle 250$ and $\displaystyle 1000$ which are exactly divisible by $\displaystyle 3$:

A person puts Rs. $\displaystyle 975$ in monthly instalment, each instalment is less than former by Rs. $\displaystyle 5$. The amount of first instalment is Rs. $\displaystyle 100$. In what time will the entire amount be paid?

If the sum of $\displaystyle n$ terms of an A.P. is $\displaystyle (3n^2-n)$ and its common difference is $\displaystyle 6$, then its first term is:

If $\displaystyle (x+1)$, $\displaystyle 3x$, $\displaystyle (4x+2)$ are in A.P. Find the value of $\displaystyle x$

Divide 144 into three parts which are in AP and such that the largest is twice the smallest, the smallest of three numbers will be:

If $\displaystyle 8^{th}$ term of an AP is 15, the sum of the 15 terms is

For what value of $\displaystyle x$, the sequence $\displaystyle x+1, 3x, 4x+2$ are in AP?

The $\displaystyle n^{th}$ term of the series whose sum to $\displaystyle n$ terms is $\displaystyle 3n^2+2n$ is

In a G.P., if the fourth term is '3' then the product of first seven terms is

$\displaystyle y = 1 + x + x^2 + \dots \infty$, then $\displaystyle x =$

Find the three numbers in G.P. whose sum is 19 and product is 216.

The sum of the 4th and 8th term of an AP is 10. Then the sum of first eleven terms of the series is

The product of three numbers which are in GP is 512. Then the second number is

Given: $\displaystyle P(7, k) = 60 P(7, k-3)$. Then: $\displaystyle k =$

If the $\displaystyle p^{th}$ term of a G.P. is $\displaystyle x$ and the $\displaystyle q^{th}$ term is $\displaystyle y$, then find the $\displaystyle n^{th}$ term:

The sum of the series $\displaystyle 0.5+0.55+0.555+\dots$ to $\displaystyle n$ terms is:

The second term of a G.P is $\displaystyle 24$ and the fifth term is $\displaystyle 81$. The series is

The series $\displaystyle 1+10^{-1}+10^{-2}+\dots$ to $\displaystyle \infty$ is

The sum of $\displaystyle m$ terms of the series $\displaystyle 1+11+111+\dots$ up to $\displaystyle m$ terms, is equal to:

If $\displaystyle (b+c-a), (c+a-b), (a+b-c)$ are in AP then $\displaystyle a, b, c$ are in:

If the AM and GM of two numbers is $\displaystyle 6.5$ and $\displaystyle 6$ the no.'s are:

If AM and HM for two numbers are $\displaystyle 3$ and $\displaystyle 3.2$, respectively. GM will be:

The $\displaystyle n^{th}$ term of the series $\displaystyle 3 + 7 + 13 + 21 + 31 + \dots$ is

The numbers $\displaystyle x$, $\displaystyle 8$, $\displaystyle y$ are in G.P. and the numbers $\displaystyle x$, $\displaystyle y$, $\displaystyle -8$ are in A.P. The values of $\displaystyle x$ and $\displaystyle y$ respectively shall be

If $\displaystyle a^x = b^y = c^z$ and $\displaystyle a, b, c$ are in G.P., the $\displaystyle x, y, z$ are in

If $\displaystyle a^x = b^y = c^z = e^{1/2}$ then $\displaystyle a, b, c$ are in GP then $\displaystyle x, y, z$ are in

The sum of the first two terms of a GP is $\displaystyle 5/3$ and the sum of infinity of the series is 3. The common ratio is

In a GP, if fourth term is $\displaystyle 3$ then the product of first seven terms is

In a G. P, sixth term is $\displaystyle 729$ and the common ratio is $\displaystyle 3$, then the first term of G.P is

In a geometric progression, the second term is $\displaystyle 12$ and sixth term is $\displaystyle 192$. Find $\displaystyle 11^{th}$ term.

If $\displaystyle 5^{th}$ term of G.P. is $\displaystyle 32$ and $\displaystyle 8^{th}$ term of G.P. is $\displaystyle 8$ then $\displaystyle 6^{th}$ term of G.P. is

Which term of the sequence $\displaystyle 2, 4, 8, 16 \dots$ is $\displaystyle 2048$?

The $\displaystyle 5^{th}$ and $\displaystyle 8^{th}$ terms of a GP series is $\displaystyle 27$ and $\displaystyle 729$. Then find the $\displaystyle 10^{th}$ term.

Four Geometric Means between $\displaystyle 4$ and $\displaystyle 972$ are

The sum up to infinity of the series $\displaystyle S = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots$ is

The $\displaystyle 3^{rd}$ term of a G.P is $\displaystyle \frac{2}{3}$ and $\displaystyle 6^{th}$ term is $\displaystyle \frac{2}{81}$, then the first term is

Find the sum to infinity of the following series: $\displaystyle 1 - 1 + 1 - 1 + \dots \infty$

Find the product of: $\displaystyle (243), (243)^{\frac{1}{6}}, (243)^{\frac{1}{36}}, \dots \infty$

The sum of the first eight terms of a G.P. is five times the sum of the first four terms; then the common ratio is:

For what values of $\displaystyle x$, the number $\displaystyle -\frac{2}{7}, x, -\frac{7}{2}$ are in G.P.?

If $\displaystyle a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}}$ and $\displaystyle a,b,c$ are in GP then $\displaystyle x, y, z$ are in

If $\displaystyle x, y$ and $\displaystyle z$ are the terms in G.P., then the term $\displaystyle x^2 + y^2, xy + yz, y^2 + z^2$ are in

The sum of the infinite series $\displaystyle 1 + \frac{2}{3} + \frac{4}{9} + \dots$ is

In a G.P. If the third term of a GP is $\displaystyle \frac{2}{3}$ and $\displaystyle 6^{th}$ term is $\displaystyle \frac{2}{81}$, then the first term is

Sum upto infinity series $\displaystyle 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n} + \dots$ is

The sum of first eight terms of geometric progression is five times the sum of the first four terms. The common ratio is

The sum of three numbers in a geometric progression is $\displaystyle 28$. When $\displaystyle 7$, $\displaystyle 2$ and $\displaystyle 1$ are subtracted from the first, second and the third numbers respectively, then the resulting numbers are in arithmetic progression. What is the sum of squares of the original three numbers?

The sum of the first two terms of an infinite geometric series is $\displaystyle 15$ and each term is equal to the sum of all the terms following it, then the sum of the series is

Three numbers are in AP and their sum is $\displaystyle 21$. If $\displaystyle 1, 5, 15$ are added to them respectively, they form a G.P. The numbers are

The sum of three numbers in G.P. is $\displaystyle 70$. If the two extremes by multiplied each by $\displaystyle 4$ and the mean by $\displaystyle 5$, the products are in AP. The numbers are

If $\displaystyle a, b, c$ are in AP and $\displaystyle x, y, z$ are in GP, then the value of $\displaystyle x^{(b-c)} y^{(c-a)} z^{(a-b)}$ is

Find the 9th term of the A.P. 8, 5, 2, -1, -4, .........

The sum of series 1 + 2 + 3 + ......... is 55. The number of terms is :

The sum of the series $\displaystyle 1 + 11 + 111 + \dots$ to $\displaystyle n$ terms is

The sum of the following series $\displaystyle 4 + 44 + 444 + \dots$ to $\displaystyle n$ terms is:

Find the sum of the series. $\displaystyle 243 + 324 + 432 + \dots$ to $\displaystyle n$ terms

The common difference of the arithmetic progression $\displaystyle \frac{1}{3}, \frac{1-3b}{3}, \frac{1-6b}{3}, \dots$ is

Find the sum to $\displaystyle n$ terms of the series: $\displaystyle 7+77+777+\dots$ to $\displaystyle n$ terms:

$\displaystyle \sum n^3$ defines:

The $\displaystyle n^{th}$ term of the sequence $\displaystyle -1, 2, -4, 8, \dots$ is

If the sum of $\displaystyle n$ terms of an A.P. is $\displaystyle 3n^2-n$ and its common difference is $\displaystyle 6$, then its third term is:

The first and last terms of an arithmetic progression are $\displaystyle 5$ and $\displaystyle 905$. Sum of the terms is $\displaystyle 45,955$. The number of terms is

A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was $\displaystyle 1,500$ after $\displaystyle 4$ years of service and $\displaystyle 1,800$ after $\displaystyle 10$ years of service, what was his starting salary and what is the annual increment in rupees?

The third term of a geometric progression is 5. Then the product of first five terms is

The sum of infinity of the series $\displaystyle \frac{1}{2} + \frac{1}{6} + \frac{1}{18} + \frac{1}{54} + \dots$ is:

A person pays $\displaystyle 975$ in monthly installments, each installment is less than former by $\displaystyle 5$. The amount of $\displaystyle 1^{st}$ installment is $\displaystyle 100$. In what time will the entire amount be paid?

A GP series consists of 2n terms. If the sum of the terms occupying the odd places is $\displaystyle S_1$ and that of the terms in even places is $\displaystyle S_2$, the common ratio of the progression is?

If the numbers $\displaystyle x, 2x + 2$ and $\displaystyle 3x + 3$ are in the geometric progression, then the fourth term of the progression is

The sum of all natural numbers between 200 and 600 those are divisible by 13 is

The sum of first two terms of a geometric progression is 14 and its infinite sum i.e. sum up to infinity is 32. What is the common ratio of the progression?