Chapter 8

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is \(9 .\) If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then (A) first term of the G.P. is \(\frac{2}{3}, 3\) (B) first term of the A.P. is \(\frac{2}{3}, 3\) (C) Common ratio of the G.P. is \(\frac{25}{2}, 2\) (D) Common difference of the A.P is \(\frac{2}{3}, 3\)

Let \(\left(1+x^{2}\right)^{2}(1+x)^{n}=\sum_{k=0} a_{k} x^{k} .\) If \(a_{1}, a_{2}, a_{3}\), are in A.P., then \(n\) is equal to (A) 1 (B) 2 (C) 3 (D) 4

If \(a, b, c\) are non-zero real numbers such that 3 \(\left(a^{2}+b^{2}+c^{2}+1\right)=2(a+b+c+a b+b c+c a)\), then, \(a, b, c\) are in (A) A.P. (B) G. P. (C) H.P. (D) all equal

Let \(t_{n}=\underbrace{1.1 \ldots 1}_{n \text { times }}\), then (A) \(t_{912}\) is not prime (B) \(t_{951}\) is not prime (C) \(t_{480}\) is not prime (D) \(t_{91}\) is not prime

Sum to \(n\) terms of the series \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots .\) is (A) \(\frac{n}{n+1}\) (B) \(\frac{2 n}{n+1}\) (C) \(\frac{n}{n-1}\) (D) None of these

Sum to infinite terms of the series \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots .\) is (A) 1 (B) 2 (C) 4 (D) None of these

Sum to infinite terms of the series \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots .\) is (A) \(\frac{1}{4}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{2}\) (D) None of thesePassage 2 A general arithmetic progression is \(a, a+d, a+2 d, \ldots\) and a general geometric progression is \(a, a r, a r^{2}, \ldots\), then the sequence \(a,(a+d) r,(a+2 d) r^{2}, \ldots\) is called an arithmetico-geometric progression (A. G.P.). Note that each term of the A.G.P. is the product of the corresponding terms of the A.P. \(a, a+d, a+2 d, \ldots\) and the G.P. \(1, r, r^{2}, \ldots\) The \(n\)th term of the A.G.P. is \([a+(n-1) d] r^{n-1}\) Sum to \(n\) terms of A.G.P. Let \(S_{n}=a+(a+d) r+(a+2 d) r^{2}+\ldots\) $$ +(a+\overline{n-2} d) r^{n-2}+(a+\overline{n-1} d) r^{n-1} $$ \(\left.\Rightarrow \quad r S_{n}=a r+(a+d) r^{2}+\ldots+\mid a+\overline{n-1} d\right) r^{n}\) Subtracting (2) from (1), we get $$ \begin{aligned} &(1-r) S_{n}=a+d r+d r^{2}+\ldots+d r^{n-1}-(a+\overline{n-1} d) r^{n} \\ &=a+\left(d r+d r^{2}+\ldots+\text { to } \overline{n-1} \text { terms }\right)-(a+\overline{n-1} d) r^{n} \\ &\left.\quad=a+\frac{d r\left(1-r^{n-1}\right)}{1-r}-\mid a+\overline{n-1} d\right) r^{n} \\ &\therefore S_{n}=\frac{a}{1-r}+\frac{d r\left(1-r^{n-1}\right)}{(1-r)^{2}}-\frac{(a+\overline{n-1} d) r^{n}}{1-r} \end{aligned} $$ \(\therefore\) Sum to infinite terms \(=S=\lim _{n \rightarrow \infty} S_{n}=\frac{a}{1-r}+\frac{d r}{(1-r)^{2}}\),

If the sum to infinity of the series \(3+(3+d) \frac{1}{4}+(3+2 d) \frac{1}{4^{2}}+\ldots\) is \(\frac{44}{9}\), then \(d=\) (A) 1 (B) 2 (C) 4 (D) None of these

\(3^{1 / 3} \cdot 9^{1 / 9} \cdot 27^{1 / 27} \cdot 81^{1 / 81} \ldots\) upto \(\infty=\) (A) \(\sqrt{27}\) (B) \(\sqrt[3]{27}\) (C) \(\sqrt[4]{27}\) (D) None of thesePassage 3 We know that arithmetic mean of the positive numbers lie between them. Suppose, \(S_{1}>S_{2}>0\) and \(S_{n+1}=\frac{1}{2}\left(S_{n}+S_{n-1}\right)\) We can easily conclude that \(S_{3}\) lies between \(S_{2}\) and \(S_{1}\) and we can write \(S_{2}

Column-I Column-II I. If the first term of an infinite G.P. is \(I\) and each term is twice the sum (A) \(\frac{2}{9}\) of the suceeding terms, then the common ratio is II. Sum to infinity of the series \(\frac{2}{3}-\frac{5}{6}+\frac{2}{3}-\frac{11}{24}+\ldots\) is (B) \(\frac{3}{2}\)III. \(\lim _{n \rightarrow \infty}\left(1+3^{-1}\right)\left(1+3^{-2}\right)\left(1+3^{-4}\right)\left(1+3^{-8}\right) \ldots\left(1+3^{-2^{\prime}}\right)=\) (C) 1 IV. If \(\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^{2}\right)=a n^{4}+b n^{3}+c n^{2}+d n+e\), then \(a+b+c+d+e=\) (D) \(\frac{1}{3}\)

Assertion: If \(a, b, c\) are distinct positive real numbers and \(a^{2}+b^{2}+c^{2}=1\), then \(a b+b c+c a\) is less than 1 . Reason: A.M. > G.M. for unequal numbers

Assertion: The value of \(x+y+z\) is 15 if \(a, x, y, z\), \(b\) are in A.P., while the value of \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\) is \(\frac{5}{3}\) if \(a, x, y, z, b\) are in H.P. The values of \(a\) and \(b\) are 9,1 respectively. Reason: The sum of \(n\) A.M.s between two quantities is equal to \(n\) times their single mean.

Assertion: For every natural number \(n,(n !)^{3} G.M. for \(n\) distinct positive quantities

If \(1, \log _{3} \sqrt{\left(3^{1-x}+2\right)}, \log _{3}(4 \cdot 3 x-1)\) are in AP, then \(x\) equals: (A) \(\log _{3} 4\) (B) \(1-\log _{3} 4\) (C) \(1-\log , 3\) (D) \(\log _{4} 3\)

The value of \(2^{1 / 4} \cdot 4^{1 / 8} \cdot 8^{1 / 16} \ldots \infty\) is: \(\quad[2002]\) (A) 1 (B) 2 (C) \(3 / 2\) (D) 4

Let \(T_{n}\) denote the number of triangles which can be formed using the vertices of a regular polygon of \(n\) sides. If \(T_{n+1}=T_{n}=21\), then \(n\) equals : [2002] (A) 5 (B) 7 (C) 6 (D) 4

The sum of the series \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\ldots \ldots \ldots\) upto $$ \infty \text { is equal to } $$ [2003] (A) \(2 \log _{e} 2\) (B) \(\log _{2} 2-1\) (C) \(\log _{e} 2\) (D) \(\log _{e}\left(\frac{4}{e}\right)\)

If \(f: R \rightarrow R\) satisfies \(f(x+y)=f(x)+f(y)\), for all \(x, y\) \(\in R\) and \(f(1)=7\), then \(\sum_{r=1}^{n} f(r)\) is [2003] (A) \(\frac{7 n}{2}\) (B) \(\frac{7(n+1)}{2}\) (C) \(7 n(n+\mathrm{l})\) (D) \(\frac{7 n(n+1)}{2}\)

If \(S_{n}=\sum_{r=0}^{n} \frac{1}{{ }^{n} C_{r}}\) and \(t_{n}=\sum_{r=0}^{n} \frac{r}{{ }^{n} C_{r}}\), then \(\frac{t_{n}}{S_{n}}\) is equal to (A) \(\frac{1}{2} n\) (B) \(\frac{1}{2} n-1\) (C) \(n-1\) (D) \(\frac{2 n-1}{2}\)

Let \(T_{r}\) be the \(r\) th term of an A.P. whose first term is \(a\) and common difference is \(d\). If for some positive integers \(m, n, m \neq n, T_{m}=\frac{1}{n}\) and \(T_{n}=\frac{1}{m}\), then \(a-d\), equals (A) 0 (B) 1 (C) \(\frac{1}{m n}\) (D) \(\frac{1}{m}+\frac{1}{n}\)

The sum of the first \(n\) terms of the series \(1^{2}+2 \cdot 2^{2}\) \(+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots\) is \(\frac{n(n+1)^{2}}{2}\) when \(n\) is even. When \(n\) is odd the sum is (A) \(\frac{3 n(n+1)}{2}\) (B) \(\frac{n^{2}(n+1)}{2}\) (C) \(\frac{n(n+1)^{2}}{4}\) (D) \(\left[\frac{n(n+1)}{2}\right]^{2}\)

\(x=\sum_{n=0}^{\infty} a^{n}, y=\sum_{n=0}^{\infty} b^{n}, z=\sum_{n=0}^{\infty} c^{n}\) where \(a, b, c\) are in A.P. and \(|a|<1,|b|<1,|c|<1\), then \(x, y, z\) are in \([2005]\) (A) G.P. (B) A.P. (C) Arithmetic - Geometric Progression (D) H.P.

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be terms of an A.P. If \(\frac{a_{1}+a_{2}+\ldots a_{p}}{a_{1}+a_{2} \ldots+a_{q}}\) \(=\frac{p^{2}}{q^{2}}, p \neq q\), then \(\frac{a_{6}}{a_{21}}\) equals \([2006]\) (A) \(\frac{41}{11}\) (B) \(\frac{7}{2}\) (C) \(\frac{2}{7}\) (D) \(\frac{11}{41}\)

If \(a_{1}, a_{2}, \ldots, a_{n}\) are in H.P., then the expression \(a_{1} a_{2}+\) \(a_{2} a_{3}+\ldots+a_{n-1} a_{n}\) is equal to [2006] (A) \(n\left(a_{1}-a_{n}\right)\) (B) \((n-1)\left(a_{1}-a_{n}\right)\) (C) \(n a_{1} a_{n}\) (D) \((n-1) a_{1} a_{n}\)

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals [2007](A) \(\frac{1}{2}(1-\sqrt{5})\) (B) \(\frac{1}{2} \sqrt{5}\) (C) \(\sqrt{5}\) (D) \(\frac{1}{2}(\sqrt{5}-1)\)

If \(p\) and \(q\) are positive real numbers such that \(p^{2}+q^{2}\) \(=1\), then the maximum value of \((p+q)\) is \(\quad\) [2007] (A) 2 (B) \(1 / 2\) (C) \(\frac{1}{\sqrt{2}}\) (D) \(\sqrt{2}\)

The first two terms of a geometric progression add up to \(12 .\) The sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is \([2008]\) (A) \(-4\) (B) \(-12\) (C) 12 (D) 4

A person is to count 4500 currency notes. Let \(a_{n}\) denote the number of notes he counts in the \(n^{\text {th }}\) minute. If \(a_{1}=a_{2}=\ldots \ldots=a_{10}=150\) and \(a_{10}, a_{11} \ldots\) are in A.P. with common difference \(-2\), then the time taken by him to count all notes is [2010] (A) 34 minutes (B) 125 minutes (C) 135 minutes (D) 24 minutes

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediate preceding month. His total saving from the start of service will be Rs. 11040 after [2011] (A) 19 months (B) 20 months (C) 21 months (D) 18 months

If 100 times the \(100^{\text {th }}\) term of an Arithmetic Progression with non zero common difference equals the 50 times its \(50^{\text {th }}\) term, then the \(150^{\text {th }}\) term of this A.P. is (A) \(-150\) (B) 150 times its \(50^{\text {th }}\) term (C) 150 (D) zero

The sum of first 20 terms of the sequence \(0.7,0.77\), \(0.777, \ldots\) is (A) \(\frac{7}{9}\left(99-10^{-20}\right)\) (B) \(\frac{7}{81}\left(179+10^{-20}\right)\) (C) \(\frac{7}{9}\left(99+10^{-20}\right)\) (D) \(\frac{7}{81}\left(179-10^{-20}\right)\)

Let \(\alpha\) and \(\beta\) be the roots of equation \(p x^{2}+q x+r-0, p \neq 0\). If \(p, q, r\) are in A.P. and \(\frac{1}{\alpha}+\frac{1}{\beta}=4\), then the value of \(|\alpha-\beta|\) is (A) \(\frac{\sqrt{61}}{9}\) (B) \(\frac{2 \sqrt{17}}{9}\) (C) \(\frac{\sqrt{34}}{9}\) (D) \(\frac{2 \sqrt{13}}{9}\)

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in \(A . P\). Then the common ratio of the G.P. is [2014] (A) \(\sqrt{2}+\sqrt{3}\) (B) \(3+\sqrt{2}\) (C) \(2-\sqrt{3}\) (D) \(2+\sqrt{3}\)

If \((10)^{9}+2(11)^{1}(10)^{8}+3(11)^{2}(10)^{7}+\ldots . .+10(11)^{9}\), \(=k(10)^{9}\) then \(k\) is equal to \(\quad\) [2014] (A) \(\frac{121}{10}\) (B) \(\frac{441}{100}\) (C) 100 (D) 110

The sum of first 9 terms of the series \(\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots \ldots\) is: \(\quad\) [2015] (A) 96 (B) 142 (C) 192 (D) 71

If the \(2^{\text {nd }}, 5^{\text {th }}\) and \(9^{\text {? }}\) terms of a non- constant A.P. are in G.P., then the common ratio of this G.P. is \([2016]\) (A) \(\frac{7}{4}\) (B) \(\frac{8}{5}\) (C) \(\frac{4}{3}\) (D) 1