Sequence & Series PYQs Set 5 (PDF)

Set 5 of 10 · Questions 121-150 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 5 (Q121-150)
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121. $If the sum of the first 20 terms of the series \frac{4 \cdot 1}{4 + 3 \cdot 1 ^{2} + 1 ^{4}} + \frac{4 \cdot 2}{4 + 3 \cdot 2 ^{2} + 2 ^{4}} + \frac{4 \cdot 3}{4 + 3 \cdot 3 ^{2} + 3 ^{4}} + \frac{4 \cdot 4}{4 + 3 \cdot 4 ^{2} + 4 ^{4}} is \frac{m}{n}, where m and n are coprime, then m + n is equal to:$ (1) $423$ (2) $421$ (3) $422$ (4) $420$ 122. $The sum of the infinite series 1 + \frac{5}{6} + \frac{12}{6 ^{2}} + \frac{22}{6 ^{3}} + \frac{35}{6 ^{4}} + \frac{51}{6 ^{5}} + \frac{70}{6 ^{6}} + .... is equal to:$ (1) $\frac{425}{216}$ (2) $\frac{429}{216}$ (3) $\frac{288}{125}$ (4) $\frac{280}{125}$ 123. $Let the sum of the first n terms of a non-constant A . P ., a_{1}, a_{2}, a_{3}, \dots \dots \dots \dots be 50 n + \frac{n ( n - 7 )}{2} A where A is aconstant. If d is the common difference of this A . P ., then the ordered pair (d, a_{50}) is equal to:$ (1) $(50, 50 + 46 A)$ (2) $(50, 50 + 45 A)$ (3) $(A, 50 + 45 A)$ (4) $(A, 50 + 46 A)$ 124. $Consider an infinite geometric series with first term ' a' and common ratio ' r'. If the sum is 4 and the second term is \frac{3}{4}, then$ (1) $a = 2, r = \frac{3}{8}$ (2) $a = \frac{4}{7}, r = \frac{3}{7}$ (3) $a = \frac{3}{2}, r = \frac{1}{2}$ (4) $a = 3, r = \frac{1}{4}$ 125. $The n^{th} term of the series 1 + 3 + 7 + 13 + 21 + \dots is 9901 . The value of n is$ (1) $90$ (2) $100$ (3) $99$ (4) $900$ 126. $If a_{n} = \frac{- 2}{4 n ^{2} - 16 n + 15}, then a_{1} + a_{2} + \dots + a_{25} is equal to$ (1) $\frac{51}{144}$ (2) $\frac{49}{138}$ (3) $\frac{50}{141}$ (4) $\frac{52}{147}$ 127. $1^{2} + (1^{2} + 2^{2}) + (1^{2} + 2^{2} + 3^{2}) + \dots + (1^{2} + 2^{2} + \dots + n^{2}) =$ (1) $\frac{n ( n + 1 ) ( n + 2 )}{12}$ (2) $\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ (3) $\frac{n ( n + 1 ) ^{2} ( n + 2 )}{12}$ (4) $\frac{n ( n + 1 ) ( n + 2 ) ( n + 3 )}{12}$ 128. $For any three positive real numbers a, b and c, 9(25 a^{2} + b^{2}) + 25(c^{2} - 3ac) = 15b(3a + c), Then$ (1) $b , c and a are in G.P.$ (2) $b, c and a are in A.P.$ (3) $a, b and c are in A.P.$ (4) $a, b and c are in G.P.$ 129. $If the sum of the first 15 terms of the (\frac{3}{4})^{3} + (1 \frac{1}{2})^{3} + (2 \frac{1}{4})^{3} + 3^{3} + (3 \frac{3}{4})^{3} + \dots \dots is equal to 225 k then k is eaual to:$ (1) $108$ (2) $27$ (3) $54$ (4) $9$ 130. $The sum of the series 1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \dots upto 11^{th} term is:$ (1) $915$ (2) $946$ (3) $945$ (4) $916$ 131. $Let a, ar, a r^{2}, be an infinite G.P. If \sum_{n = 0}^{\infty} a r^{n} = 57 and \sum_{n = 0}^{\infty} a^{3} r^{3 n} = 9747, then a + 18 r is equal to$ (1) $46$ (2) $38$ (3) $31$ (4) $27$ 132. $If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:$ (1) $757$ (2) $755$ (3) $750$ (4) $760$ 133. $Let a, b and c be in G.P. with common ratio r, where a \neq = 0 and 0 < r < \frac{1}{2} . If 3 a, 7 b and 15 c are the first three terms of an A.P., then the 4^{t h} term of this A.P. is :$ (1) $\frac{2}{3} a$ (2) $5 a$ (3) $\frac{7}{3} a$ (4) $a$ 134. $The sum of the following series 1 + 6 + \frac{9 ( 1 ^{2} + 2 ^{2} + 3 ^{2} )}{7} + \frac{12 ( 1 ^{2} + 2 ^{2} + 3 ^{2} + 4 ^{2} )}{9} + \frac{15 ( 1 ^{2} + 2 ^{2} + \dots . + 5 ^{2} )}{11} + \dots up to 15 terms, is:$ (1) $7520$ (2) $7510$ (3) $7830$ (4) $7820$ 135. $If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \dots is (102) m, then m is equal to:$ (1) $20$ (2) $25$ (3) $10$ (4) $5$ 136. $Let the greatest common divisor of m, n be 1 . If \frac{1}{1.7} + \frac{1}{7.13} + \frac{1}{13.19} + \dots . upto 20 terms = \frac{m}{n}, then 5 m + 2 n =$ (1) $325$ (2) $330$ (3) $342$ (4) $337$ 137. $If \sum_{r = 1}^{n} T_{r} = \frac{( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 )}{64}, then lim_{n \to \infty} \sum_{r = 1}^{n} (\frac{1}{T _{r}}) is equal to:$ (1) $0$ (2) $\frac{2}{3}$ (3) $1$ (4) $\frac{1}{3}$ 138. $Let a_{1}, a_{2}, \dots, a_{10} be a G . P . If \frac{a _{3}}{a _{1}} = 25, then \frac{a _{9}}{a _{5}} equals :$ (1) $5^{4}$ (2) $4 (5^{2})$ (3) $5^{3}$ (4) $2 (5^{2})$ 139. $Let a_{n} be the n^{th} term of an A.P. If S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = 700, a_{6} = 7 and S_{7} = 7, then a_{n} is equal to:$ (1) $65$ (2) $56$ (3) $70$ (4) $64$ 140. $If f : R \to R is defined by f (x + y) = f (x) + f (y) \forall x, y \in R and f (1) = 7, then \sum_{r = 1}^{n} f (r) =$ (1) $\frac{3 n ( n + 2 )}{4}$ (2) $\frac{n ( n - 1 )}{2}$ (3) $\frac{7 n ( n + 1 )}{2}$ (4) $\frac{( n + 1 ) ( n + 2 )}{4}$ 141. $The product of three consecutive terms of a G . P . is 512 . If 4 is added to each of the first and the second of these terms, the three terms now form an A . P . Then the sum of the original three terms of the given G . P . is:$ (1) $36$ (2) $32$ (3) $24$ (4) $28$ 142. $Let a_{1}, a_{2}, \dots a_{n} be a given A . P . whose common difference is an integer and S_{n} = a_{1} + a_{2} + \dots + a_{n} If a_{1} = 1, a_{n} = 300 and 15 \leq n \leq 50 then the ordered pair (S_{n - 4}, a_{n - 4}) is equal to$ (1) $(2490, 249)$ (2) $(2480, 249)$ (3) $(2490, 248)$ (4) $(2480, 248)$ 143. $The interior angles of a polygon with n sides, are in an A.P. with common difference 6^{\circ} . If the largest interior angle of the polygon is 219^{\circ}, then n is equal to$ 144. $The value of (0.16)^{log_{2.5} (\frac{1}{3} + \frac{1}{3 ^{2}} + \frac{1}{3 ^{3}} + ... of \infty)} is equal to$ 145. $If three successive terms of a G.P. with common ratio r (r > 1) are the lengths of the sides of a triangle and [r] denotes the greatest integer less than or equal to r, then 3 [r] + [- r] is equal to :$ 146. $Let the first term of a series be T_{1} = 6 and its r^{th} term T_{r} = 3 T_{r - 1} + 6^{r}, r = 2, 3, n . If the sum of the first n terms of this series is \frac{1}{5} (n^{2} - 12 n + 39) (4 \cdot 6^{n} - 5 \cdot 3^{n} + 1), then n is equal to______$ 147. $Let a_{1} = b_{2} = 1 and a_{n} = a_{n - 1} + (n - 1), b_{n} = b_{n - 1} + a_{n - 1} \forall n \geq 2 . If S = \sum \frac{b _{n}}{2 ^{n}} and T = \sum \frac{n}{2 ^{n - 1}}, then 2^{7} (2 S - T) is equal to ______$ 148. $If (\frac{1}{α + 1} + \frac{1}{α + 2} + \dots \dots + \frac{1}{α + 1012}) - (\frac{1}{2 \cdot 1} + \frac{1}{4 \cdot 3} + \frac{1}{6 \cdot 5} + \dots \dots + \frac{1}{2024 \cdot 2023}) = \frac{1}{2024}, then α is equal to______$ 149. $If \frac{1 ^{3} + 2 ^{3} + 3 ^{3} + \dots up to n terms}{1.3 + 2.5 + 3.7 + \dots up to n terms} = \frac{9}{5}, then the value of n is$ 150. $If 1 + \frac{3 - 2}{2 3} + \frac{5 - 2 6}{18} + \frac{9 3 - 11 2}{36 3} + \frac{49 - 20 6}{180} + \dots upto \infty = 2 + (\frac{b}{a} + 1) log_{e} (\frac{a}{b}), where a and b are integers with gcd (a, b) = 1, then 11 a + 18 b is equal to_______$