Sequences and Series

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum\limits_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=...

The interior angles of a polygon with n sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then n is equal to _______.

For positive integers $n$, if $4 a_n=\left(n^2+5 n+6\right)$ and $S_n=\sum\limits_{k=1}^n\left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is :

Let $S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $\sqrt{2026 \mathrm{~S}_{2025}}$, then the absolute difference bet...

If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\frac{1}{7^3}(5+3 \alpha)+\ldots \ldots \ldots \ldots \infty$, then the value of $\alpha$ is :

In an arithmetic progression, if $\mathrm{S}_{40}=1030$ and $\mathrm{S}_{12}=57$, then $\mathrm{S}_{30}-\mathrm{S}_{10}$ is equal to :

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

The roots of the quadratic equation $3 x^2-p x+q=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88...

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

Suppose that the number of terms in an A.P. is $2 k, k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to:

If the sum of the series \(\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}\) is equal to 5, then \(50 \mathrm{~d}\) is equal to :

If \(\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots . .+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots \ldots+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}\), then \(\alpha\) is e...

Let \(a, a r, a r^2\), ............ be an infinite G.P. If \(\sum_\limits{n=0}^{\infty} a r^n=57\) and \(\sum_\limits{n=0}^{\infty} a^3 r^{3 n}=9747\), then \(a+18 r\) is equal to

Let the positive integers be written in the form :

If the \(k^{\text {th }}\) row contains exactly \(k\) numbers for every natural number \(k\), then the row in which the number 5310 will be, is __________.

Let \(\alpha=\sum_\limits{r=0}^n\left(4 r^2+2 r+1\right){ }^n C_r\) and \(\beta=\left(\sum_\limits{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}\). If \(140<\frac{2 \alpha}{\beta}<281\), then the value of \(n\) is _________.

An arithmetic progression is written in the following way

The sum of all the terms of the 10th row is _________.

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \(\frac{70}{3}\) and the product of the third and fifth terms is 49. Then the sum of the \(4^{\text {th }}, 6^{\text {th }}\) and $$8^{\text ...

Let the first term of a series be \(T_1=6\) and its \(r^{\text {th }}\) term \(T_r=3 T_{r-1}+6^r, r=2,3\),
............ \(n\). If the sum of the first \(n\) terms of this series is $$\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \c...

If \(\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0\), and \((60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}\), where \(a, b \in N\), then \((a+b)\) equal to _________.

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took ...

Let \(A B C\) be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle \(A B C\) and the same process is repeated infinitely many times. If \(\mathrm{P}\) is the sum of perimeters and $$...

Let \(a_1, a_2, a_3, \ldots\) be in an arithmetic progression of positive terms.
Let \(A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2 k-1}^2-a_{2 k}^2\).
If \(\mathrm{A}_3=-153, \mathrm{~A}_5=-435\) and $$\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3...

If \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m\) and \(\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=\mathrm{n}\), then the point \((\mathrm{m}, \mathrm{n})\) lie...

If \(1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots\) upto \(\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)\), where a ...

For \(x \geqslant 0\), the least value of \(\mathrm{K}\), for which \(4^{1+x}+4^{1-x}, \frac{\mathrm{K}}{2}, 16^x+16^{-x}\) are three consecutive terms of an A.P., is equal to :

Let the first three terms 2, p and q, with \(q \neq 2\), of a G.P. be respectively the \(7^{\text {th }}, 8^{\text {th }}\) and \(13^{\text {th }}\) terms of an A.P. If the \(5^{\text {th }}\) term of the G.P. is the \(n^{\text {th }}\) ter...

Let three real numbers \(a, b, c\) be in arithmetic progression and \(a+1, b, c+3\) be in geometric progression. If \(a>10\) and the arithmetic mean of \(a, b\) and \(c\) is 8, then the cube of the geometric mean of \(a, b\) and \(c\) is

The value of \(\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots .+100^2 \times 101}\) is

The sum of the series \(\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots\) up to 10 -terms is

For \(0 < c < b < a\), let \((a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements
(I) If \(\alpha \in(-1,0)\), then \(b\) cannot be the geometric mean of $a$ and \(c\)
(II) If $$\a...

Let \(2^{\text {nd }}, 8^{\text {th }}\) and \(44^{\text {th }}\) terms of a non-constant A. P. be respectively the \(1^{\text {st }}, 2^{\text {nd }}\) and \(3^{\text {rd }}\) terms of a G. P. If the first term of the A. P. is 1, then the ...

Let \(\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots\) upto 10 terms and \(\beta=\sum_\limits{n=1}^{10} n^4\). If \(4 \alpha-\beta=55 k+40\), then \(\mathrm{k}\) is equal to __________.

Let \(S_n\) denote the sum of first \(n\) terms of an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(\mathrm{S}_{15}-\mathrm{S}_5\) is :

Let \(S_n\) be the sum to \(n\)-terms of an arithmetic progression \(3,7,11\),
If \(40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42\), then \(n\) equals ________.

Let \(a\) and \(b\) be be two distinct positive real numbers. Let \(11^{\text {th }}\) term of a GP, whose first term is \(a\) and third term is \(b\), is equal to \(p^{\text {th }}\) term of another GP, whose first term is \(a\) and fifth ...

In an A.P., the sixth term \(a_6=2\). If the product \(a_1 a_4 a_5\) is the greatest, then the common difference of the A.P. is equal to

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

If each term of a geometric progression \(a_1, a_2, a_3, \ldots\) with \(a_1=\frac{1}{8}\) and \(a_2 \neq a_1\), is the arithmetic mean of the next two terms and \(S_n=a_1+a_2+\ldots . .+a_n\), then \(S_{20}-S_{18}\) is equal to

If \(\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}\) are in an A.P. and \(\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e\) a are also in an A.P, then $$a: b:...

If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \cdots \infty$, then the value of $p$ is ____________.

The number of common terms in the progressions $4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and $3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :

\(\text { The } 20^{\text {th }} \text { term from the end of the progression } 20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4} \text { is : }\)

Let $3,7,11,15, \ldots, 403$ and $2,5,8,11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ___________.

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

If three successive terms of a G.P. with common ratio $\mathrm{r}(\mathrm{r}>1)$ are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r]+[-r]$ is equal to _____________.

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7$, then $\mathrm{S}_{15}-\mathrm{S}_5$ is equal to :

Let \(S_{K}=\frac{1+2+\ldots+K}{K}\) and \(\sum_\limits{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right)\), where \(A, B, C, D \in \mathbb{N}\) and \(A\) has least value. Then

Let \(0 < z < y < x\) be three real numbers such that \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in an arithmetic progression and \(x, \sqrt{2} y, z\) are in a geometric progression. If \(x y+y z+z x=\frac{3}{\sqrt{2}} x y z\) , then $$3...

Let \(\mathrm{a}_{\mathrm{n}}\) be the \(\mathrm{n}^{\text {th }}\) term of the series \(5+8+14+23+35+50+\ldots\) and \(\mathrm{S}_{\mathrm{n}}=\sum_\limits{k=1}^{n} a_{k}\). Then \(\mathrm{S}_{30}-a_{40}\) is equal to :

The sum of the first \(20\) terms of the series \(5+11+19+29+41+\ldots\) is :