Chapter 10

Assertion: If \(a, b, c, d \in R+\) and \(a, b, c, d\) are in H.P., then \(b+c>a+d\) Reason: H.M > A.M. for unequal numbers

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

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

Fifth term of a GP is 2, then the product of its 9 terms is : \(\quad[2002]\) (A) 256 (B) 512 (C) 1024 (D) None of these

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) \(7 n(n+1)\) (D) \(\frac{7 n(n+1)}{2}\)

The sum of the first \(\mathrm{n}\) terms of the series \(\mathrm{I}^{2}+2 \cdot 2^{2}\) \(+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots\) is \(\frac{n(n+1)^{2}}{2}\) when \(n\) is even. When \(n\) is odd the sum is \(\quad\) [2004] (A) \(\frac{3 n(n+1)}{2}\) (B) \(\frac{n^{2}(n+1)}{2}\) (C) \(\frac{n(n+1)^{2}}{4}\) (D) \(\left[\frac{n(n+1)}{2}\right]^{2}\)

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 \(\quad\) [2006] (A) \(\frac{41}{11}\) (B) \(\frac{7}{2}\) (C) \(\frac{2}{7}\) (D) \(\frac{11}{41}\)

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 (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 \(\underline{\text { [2007 }}\) (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 (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 \(|\mathbf{2 0 1 1}|\) (A) 19 months (B) 20 months (C) 21 months (D) 18 months

Statement 1: The sum of the series \(1+(1+2+4)+\) \((4+6+9)+(9+12+16)+\ldots . .+(361+380+400)\) is 8000 . Statement 2: \(\sum_{k=1}^{n}\left(k^{3}=(k-1)^{3}\right)=n^{3}\) for any natural number \(n .\) (A) Statement 1 is false, statement 2 is true (B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1 (C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1 (D) Statement 1 is true, statement 2 is false

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 |2012| (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) \(\frac{7}{81}\left(179-10^{-20}\right)\)

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 (A) \(\sqrt{2}+\sqrt{3}\) \(\mid 2014]\) (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 \(\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots . .\) is: (A) 96 (B) 142 (C) 192 (D) 71

If the \(2^{\text {nd }}, 5^{\text {th }}\) and \(9^{\text {th }}\) 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

If the sum of the first terms of the series \(\left(1 \frac{3}{5}\right)^{2}+\left(2 \frac{2}{5}\right)^{2}+\left(3 \frac{1}{5}\right)^{2}+4^{2}+\left(4 \frac{4}{5}\right)^{2}+\ldots\), is \(\frac{16}{5} \mathrm{~m}\) then \(m\) is equal to [2016] (A) 99 (B) 102 (C) 101 (D) 100