JEE Mathematics Progressions Related Inequalities and Series MCQs Set 04

Choose the most appropriate option (a, b, c or d).

Question. \( \sum_{r=1}^{n} r^2 - \sum_{m=1}^{n} \sum_{r=1}^{m} r \) is equal to
(a) 0
(b) \( \frac{1}{2} \left( \sum_{r=1}^n r^2 + \sum_{r=1}^n r \right) \)
(c) \( \frac{1}{2} \left( \sum_{r=1}^n r^2 - \sum_{r=1}^n r \right) \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \left( \sum_{r=1}^n r^2 - \sum_{r=1}^n r \right) \)

Question. If \( (1 + x)(1 + x^2)(1 + x^4)\dots(1 + x^{128}) = \sum_{r=0}^n x^r \) then n is
(a) 255
(b) 127
(c) 63
(d) None of the options
Answer: (a) 255

Question. The value of \( \sum_{n=1}^{m} \log \frac{a^{2n-1}}{b^{m-1}} \) (\( a \neq 0, 1; b \neq 0, 1 \)) is
(a) \( m \log \frac{a^{2m}}{b^{m-1}} \)
(b) \( \log \frac{a^{2m}}{b^{m-1}} \)
(c) \( \frac{m}{2} \log \frac{a^{2m}}{b^{2m-2}} \)
(d) \( \frac{m}{2} \log \frac{a^{2m}}{b^{m+1}} \)
Answer: (c) \( \frac{m}{2} \log \frac{a^{2m}}{b^{2m-2}} \)

Question. The sum of the products of the ten numbers \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 5 \) taking two at a time is
(a) 165
(b) -55
(c) 55
(d) None of the options
Answer: (b) -55

Question. The sum of the series \( \frac{1}{\log_2 4} + \frac{1}{\log_4 4} + \frac{1}{\log_8 4} + \dots + \frac{1}{\log_{2^n} 4} \) is
(a) \( \frac{n(n + 1)}{2} \)
(b) \( \frac{n(n + 1)(2n + 1)}{12} \)
(c) \( \frac{1}{n(n + 1)} \)
(d) None of the options
Answer: (d) None of the options

Question. If \( \sum_{n=1}^n n, \frac{\sqrt{10}}{3} \sum_{n=1}^n n^2, \sum_{n=1}^n n^3 \) are in GP then the value of n is
(a) 2
(b) 3
(c) 4
(d) nonexistent
Answer: (c) 4

Question. The value of \( \sum_{r=1}^{n} \{(2r - 1)a + \frac{1}{b^r}\} \) is equal to
(a) \( an^2 + \frac{b^{n-1} - 1}{b^{n-1}(b - 1)} \)
(b) \( an^2 + \frac{b^n - 1}{b^n(b - 1)} \)
(c) \( an^3 + \frac{b^{n-1} - 1}{b^n(n - 1)} \)
(d) None of the options
Answer: (b) \( an^2 + \frac{b^n - 1}{b^n(b - 1)} \)

Question. If \( s_n = \sum_{n=1}^{n} \frac{1 + 2 + 2^2 + \dots \text{to n terms}}{2^n} \) the \( s_n \) is equal to
(a) \( 2^n - (n + 1) \)
(b) \( 1 - \frac{1}{2^n} \)
(c) \( n - 1 + \frac{1}{2^n} \)
(d) \( 2^n - 1 \)
Answer: (c) \( n - 1 + \frac{1}{2^n} \)

Question. Let \( S_n \) denote the sum of the cubes of the first n natural numbers \( s_n \) denote the sum of the first n natural numbers. Then \( \sum_{r=1}^{n} \frac{S_r}{s_r} \) is equal to
(a) \( \frac{n(n + 1)(n + 2)}{6} \)
(b) \( \frac{n(n + 1)}{2} \)
(c) \( \frac{n^2 + 3n + 2}{6} \)
(d) None of the options
Answer: (a) \( \frac{n(n + 1)(n + 2)}{6} \)

Question. It is known that \( \sum_{r=1}^{\infty} \frac{1}{(2r - 1)^2} = \frac{\pi^2}{8} \). Then \( \sum_{r=1}^{\infty} \frac{1}{r^2} \) is equal to
(a) \( \frac{\pi^2}{24} \)
(b) \( \frac{\pi^2}{3} \)
(c) \( \frac{\pi^2}{6} \)
(d) None of the options
Answer: (c) \( \frac{\pi^2}{6} \)

Question. It is given that \( \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \dots \text{to } \infty = \frac{\pi^4}{90} \). Then \( \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \dots \text{to } \infty \) is equal to
(a) \( \frac{\pi^4}{96} \)
(b) \( \frac{\pi^4}{45} \)
(c) \( \frac{89\pi^4}{90} \)
(d) None of the options
Answer: (a) \( \frac{\pi^4}{96} \)

Question. If in a series \( t_n = \frac{n}{(n + 1)!} \) then \( \sum_{n=1}^{20} t_n \) is equal to
(a) \( \frac{20! - 1}{20!} \)
(b) \( \frac{21! - 1}{21!} \)
(c) \( \frac{1}{2(n - 1)!} \)
(d) None of the options
Answer: (b) \( \frac{21! - 1}{21!} \)

Question. If \( t_n \) denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + …. Then \( t_{50} \) is
(a) \( 49^2 - 1 \)
(b) \( 49^2 + 2 \)
(c) \( 50^2 + 1 \)
(d) \( 49^2 + 2 \)
Answer: (d) \( 49^2 + 2 \)

Question. \( 2^{1/4} \cdot 4^{1/8} \cdot 8^{1/16} \dots \text{to } \infty \) is equal to
(a) 1
(b) 2
(c) \( \frac{3}{2} \)
(d) None of the options
Answer: (b) 2

Question. The sum of n terms of the series \( 1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \dots \) is \( \frac{n(n + 1)^2}{2} \) when n is even. When n is odd, the sum is
(a) \( \frac{n^2(n + 1)}{2} \)
(b) \( \frac{n^2(n - 1)}{2} \)
(c) \( 2(n + 1)^2 \cdot (2n + 1) \)
(d) None of the options
Answer: (a) \( \frac{n^2(n + 1)}{2} \)

Question. If n is an odd integer greater than or equal to 1 then the value of \( n^3 - (n - 1)^3 + (n - 2)^3 - \dots + (-1)^{n-1} \cdot 1^3 \) is
(a) \( \frac{(n + 1)^2 \cdot (2n - 1)}{4} \)
(b) \( \frac{(n - 1)^2 \cdot (2n - 1)}{4} \)
(c) \( \frac{(n + 1)^2 \cdot (2n + 1)}{4} \)
(d) None of the options
Answer: (c) \( \frac{(n + 1)^2 \cdot (2n + 1)}{4} \)

Question. Observe that \( 1^3 = 1, 2^3 = 3 + 5, 3^3 = 7 + 9 + 11, 4^3 = 13 + 15 + 17 + 19 \). Then \( n^3 \) as a similar series is
(a) \( \left\{ 2 \left[ \frac{n(n - 1)}{2} + 1 \right] - 1 \right\} + \left\{ 2 \left[ \frac{(n + 1)n}{2} + 1 \right] + 1 \right\} + \dots + \left\{ 2 \left[ \frac{(n + 1)n}{2} + 1 \right] + 2n - 3 \right\} \)
(b) \( (n^2 + n + 1) + (n^2 + n + 3) + (n^2 + n + 5) + \dots + (n^2 + 3n - 1) \)
(c) \( (n^2 - n + 1) + (n^2 - n + 3) + (n^2 - n + 5) + \dots + (n^2 + n - 1) \)
(d) None of the options
Answer: (c) \( (n^2 - n + 1) + (n^2 - n + 3) + (n^2 - n + 5) + \dots + (n^2 + n - 1) \)

Question. Let \( t_r = 2^{r/2} + 2^{-r/2} \). Then \( \sum_{r=1}^{10} t_r^2 \) is equal to
(a) \( \frac{2^{11} - 1}{2^{10}} + 20 \)
(b) \( \frac{2^{11} - 1}{2^{10}} + 19 \)
(c) \( \frac{2^{21} - 1}{2^{20}} - 1 \)
(d) None of the options
Answer: (b) \( \frac{2^{11} - 1}{2^{10}} + 19 \)

Question. Let \( S_k = \lim_{n \to \infty} \sum_{i=0}^n \frac{1}{(k + 1)^i} \). Then \( \sum_{k=1}^n kS_k \) equals
(a) \( \frac{n(n + 1)}{2} \)
(b) \( \frac{n(n - 1)}{2} \)
(c) \( \frac{n(n + 2)}{2} \)
(d) \( \frac{n(n + 3)}{2} \)
Answer: (d) \( \frac{n(n + 3)}{2} \)

Question. Let \( t_n = n \cdot (n!) \). then \( \sum_{n=1}^{15} t_n \) is equal to
(a) 15! – 1
(b) 15! + 1
(c) 16! – 1
(d) None of the options
Answer: (c) 16! – 1

Question. The sum of \( \frac{3}{1 \cdot 2} \cdot \frac{1}{2} + \frac{4}{2 \cdot 3} \cdot \left(\frac{1}{2}\right)^2 + \frac{5}{3 \cdot 4} \cdot \left(\frac{1}{2}\right)^3 + \dots \text{to n terms is equal to} \)
(a) \( 1 - \frac{1}{(n + 1)2^n} \)
(b) \( 1 - \frac{1}{n \cdot 2^{n-1}} \)
(c) \( 1 + \frac{1}{(n + 1)2^n} \)
(d) None of the options
Answer: (a) \( 1 - \frac{1}{(n + 1)2^n} \)

Question. Let \( f(n) = \left[ \frac{1}{2} + \frac{n}{100} \right] \) where [x] denotes the integral part of x. Then the value of \( \sum_{n=1}^{100} f(n) \) is
(a) 50
(b) 51
(c) 1
(d) None of the options
Answer: (b) 51

Question. \( A_r; r = 1, 2, 3, \dots, n \) are n points on the parabola \( y^2 = 4x \) in the first quadrant. If \( A_r = (x_r, y_r) \), where \( x_1, x_2, x_3, \dots, x_n \) are in GP and \( x_1 = 1, x_2 = 2 \), then \( y_n \) is equal to
(a) \( -2^{\frac{n+1}{2}} \)
(b) \( 2^{n+1} \)
(c) \( (\sqrt{2})^{n+1} \)
(d) \( 2^{\frac{n}{2}} \)
Answer: (c) \( (\sqrt{2})^{n+1} \)

Question. In the given square, a diagonal is drawn, and parallel line the segments joining points on the adjacent sides are drawn on both sides of the diagonal. The length of the diagonal \( n\sqrt{2} \) cm. If the distance between consecutive line segment be \( 1/\sqrt{2} \) cm then the sum of the lengths of all possible line segments and the diagonal is
(a) \( n(n + 1)\sqrt{2} \) cm
(b) \( n^2 \) cm
(c) \( n(n + 2) \) cm
(d) \( n^2 \sqrt{2} \) cm
Answer: (d) \( n^2 \sqrt{2} \) cm

Question. ABCD is a square of length a, \( a \in N, a > 1 \). Let \( L_1, L_2, L_3, \dots \) be points on BC such that \( BL_1 = L_1L_2 = L_2L_3 = \dots = 1 \) and \( M_1, M_2, M_3, \dots \) be points on CD such that \( CM_1 = M_1M_2 = M_2M_3 = \dots = 1 \). Then \( \sum_{n=1}^{a-1} (AL_n^2 + L_n M_n^2) \) is equal to
(a) \( \frac{1}{2} a(a - 1)^2 \)
(b) \( \frac{1}{2} a(a - 1)(4a - 1) \)
(c) \( \frac{1}{2} (a - 1)(2a - 1)(4a - 1) \)
(d) None of the options
Answer: (b) \( \frac{1}{2} a(a - 1)(4a - 1) \)

Question. The sum of infinite terms of a decreasing GP is equal to the greatest value of the function \( f(x) = x^3 + 3x - 9 \) in the interval [-2, 3] and the difference between the first two terms is f'(0). Then the common ratio of the GP is
(a) \( -\frac{2}{3} \)
(b) \( \frac{4}{3} \)
(c) \( \frac{2}{3} \)
(d) \( -\frac{4}{3} \)
Answer: (c) \( \frac{2}{3} \)

Question. The lengths of three unequal edges of a rectangular solid block are in GP. The volume of the block is \( 216 \text{ cm}^3 \) and the total surface area is \( 252 \text{ cm}^2 \). The length of the longest edge is
(a) 12 cm
(b) 6 cm
(c) 18 cm
(d) 3 cm
Answer: (a) 12 cm

Question. ABC is a right-angled triangle in which \( \angle B = 90^\circ \) and BC = a. If n points \( L_1, L_2, \dots, L_n \) on AB are such that AB is divided in \( n + 1 \) equal parts and \( L_1M_1, L_2M_2, \dots, L_n M_n \) are line segments parallel to BC and \( M_1, M_2, \dots, M_n \) are on AC then the sum of the lengths of \( L_1M_1, L_2M_2, \dots, L_n M_n \) is
(a) \( \frac{a(n + 1)}{2} \)
(b) \( \frac{a(n - 1)}{2} \)
(c) \( \frac{an}{2} \)
(d) impossible to find from the given data
Answer: (c) \( \frac{an}{2} \)

Choose the correct options. One or more options may be correct.

Question. Three positive numbers from a GP. If the middle number is increased by 8, the three numbers form an AP. If the last number is also increased by 64 along with the previous increase in the middle number, the resulting numbers from a GP again. Then
(a) common ratio = 3
(b) first number = \( \frac{4}{9} \)
(c) common ratio = -5
(d) first number = 4
Answer: (a) common ratio = 3

Question. If \( a, b, c \) are in GP and \( a, p, q \) are in AP such that \( 2a, b + p, c + q \) are in GP then the common difference of the AP is
(a) \( \sqrt{2} a \)
(b) \( (\sqrt{2} + 1)(a - b) \)
(c) \( \sqrt{2}(a + b) \)
(d) \( (\sqrt{2} - 1)(b - a) \)
Answer: (b) \( (\sqrt{2} + 1)(a - b) \)

Question. If \( x, y, z \) are positive numbers in AP then
(a) \( y^2 \geq xy \)
(b) \( y \geq 2\sqrt{xz} \)
(c) \( \frac{x + y}{2y - x} + \frac{y + z}{2y - z} \) has the minimum value 2
(d) \( \frac{x + y}{2y - x} + \frac{y + z}{2y - z} \geq 4 \)
Answer: (a) \( y^2 \geq xy \)

Question. Between two unequal numbers, if \( a_1, a_2 \) are two AMs; \( g_1, g_2 \) are two GMs and \( h_1, h_2 \) are two HMs then \( g_1 \cdot g_2 \) is equal to
(a) \( a_1 h_1 \)
(b) \( a_1 h_2 \)
(c) \( a_2 h_2 \)
(d) \( a_2 h_1 \)
Answer: (b) \( a_1 h_2 \)

Question. The number 1, 4, 16 can be three terms (not necessarily consecutive) of
(a) no AP
(b) only one GP
(c) infinite number of APs
(d) infinite number of GPs
Answer: (c) infinite number of APs