Sequence & Series PYQs Set 6 (PDF)

Set 6 of 10 · Questions 151-180 of 284

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 6 (Q151-180)
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151. $Let a_{1}, a_{2}, a_{3}, \dots be a G.P. of increasing positive terms. If a_{1} a_{5} = 28 and a_{2} + a_{4} = 29, then a_{6} is equal to:$ (1) $628$ (2) $812$ (3) $526$ (4) $784$ 152. $Let P (n) : 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{6 ( n - 1 ) ( n - 2 ) \dots ( n - 2020 ) + 2 n ^{3} + 3 n ^{2} + n}{6}, for all n \in N . Then which of the following is correct?$ (1) $P (n) is true for all n \in N$ (2) $P(n) is true for all h > 2020$ (3) $P(n) is true for all n\leq2020$ (4) $P (n) is not true for any n \in N$ 153. $Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to$ (1) $241$ (2) $231$ (3) $210$ (4) $220$ 154. $If the middle term of the Λ . P . is 300 , then the sum of its first 51 terms is$ (1) $15300$ (2) $14800$ (3) $16500$ (4) $14300$ 155. $The sum 1 + 3 + 11 + 25 + 45 + 71 + \dots upto 20 terms, is equal to$ (1) $7240$ (2) $8124$ (3) $7130$ (4) $6982$ 156. $For any integer n \geq 1, \sum_{K = 1}^{n} K (K + 2) =$ (1) $\frac{n ( n + 1 ) ( n + 2 )}{6}$ (2) $\frac{n ( n + 1 ) ( 2 n + 7 )}{6}$ (3) $\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ (4) $\frac{n ( n - 1 ) ( 2 n + 8 )}{6}$ 157. $For all n \in N, \frac{1}{2 \cdot 5} + \frac{1}{5 \cdot 8} + \dots + \frac{1}{( 3 n - 1 ) ( 3 n + 2 )} is equal to$ (1) $\frac{n}{6 n + 4}$ (2) $\frac{n ^{2}}{6 n + 4}$ (3) $\frac{1}{2} \cdot \frac{n ^{2}}{6 n + 4}$ (4) $\frac{n}{6 n ^{2} + 4}$ 158. $Let a and b be be two distinct positive real numbers. Let 11^{th} term of a GP, whose first term is a and third term is b, is equal to p^{th} term of another GP, whose first term is a and fifth term is b . Then p is equal to$ (1) $20$ (2) $25$ (3) $21$ (4) $24$ 159. $The sum 1 + \frac{1 + 3}{2 !} + \frac{1 + 3 + 5}{3 !} + \frac{1 + 3 + 5 + 7}{4 !} + \dots upto \infty terms, is equal to$ (1) $3 e$ (2) $2 e$ (3) $4 e$ (4) $6 e$ 160. $The sum to 10 terms of the series \frac{1}{1 + 1 ^{2} + 1 ^{4}} + \frac{2}{1 + 2 ^{2} + 2 ^{4}} + \frac{3}{1 + 3 ^{2} + 3 ^{4}} + .... is$ (1) $\frac{59}{111}$ (2) $\frac{55}{111}$ (3) $\frac{56}{111}$ (4) $\frac{58}{111}$ 161. $Let 729, 81, 9, 1, \dots be a sequence and P_{n} denote the product of the first n terms of this sequence. If 2 \sum (P_{n})^{\frac{1}{n}} = \frac{3 ^{α} - 1}{3 ^{β}} and gcd (α, β) = 1, then α + β is equal to$ (1) $74$ (2) $76$ (3) $73$ (4) $75$ 162. $If^{n} C_{4},^{n} C_{5} and^{n} C_{6} are in A . P ., then n can be$ (1) $9$ (2) $14$ (3) $11$ (4) $12$ 163. $The number of common terms in the progressions 4, 9, 14, 19, \dots \dots, up to 25^{th} term and 3, 6, 9, 12, up to 37^{th} term is :$ (1) $9$ (2) $5$ (3) $7$ (4) $8$ 164. $If a_{1}, a_{2}, a_{3}, \dots .. a_{n} are in A . P . and a_{1} + a_{4} + a_{7} + \dots + a_{16} = 114, then a_{1} + a_{6} + a_{11} + a_{16} is equal to :$ (1) $98$ (2) $76$ (3) $38$ (4) $64$ 165. $Let a, b and c be the 7^{th}, 11^{th} and 13^{th} terms respectively of a non-constant A . P . If these are also the three consecutive terms of a G . P ., then \frac{a}{c} is equal to:$ (1) $2$ (2) $\frac{1}{2}$ (3) $\frac{7}{13}$ (4) $4$ 166. $Let a_{1}, a_{2}, a_{3}, \dots be G.P. of increasing positive terms such that a_{2} \cdot a_{3} \cdot a_{4} = 64 and a_{1} + a_{3} + a_{5} = \frac{813}{7} . Then a_{3} + a_{5} + a_{7} is equal to:$ (1) $3244$ (2) $3248$ (3) $3252$ (4) $3256$ 167. $The sum of the series 1 + \frac{2}{3} (\frac{1}{8}) + \frac{2 \times 5}{3 \times 6} (\frac{1}{8})^{2} + \frac{2 \times 5 \times 8}{3 \times 6 \times 9} (\frac{1}{8})^{3} + \dots is$ (1) $\frac{4}{3 49}$ (2) $\frac{3 49}{4}$ (3) $\frac{4}{3 81}$ (4) $\frac{3 81}{4}$ 168. $If 19^{th} term of a non-zero A . P . is zero, then its (49^{th} term): (29^{th} term) is:$ (1) $4 : 1$ (2) $1 : 3$ (3) $3 : 1$ (4) $2 : 1$ 169. $The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by \frac{21}{2} . Then the number of terms which are integers in the A.P. is:$ (1) $6$ (2) $4$ (3) $8$ (4) $10$ 170. $If a_{1}, a_{2}, a_{3}, \dots, a_{10} is a geometric progression and \frac{a _{3}}{a _{1}} = 25, then \frac{a _{9}}{a _{5}} equals$ (1) $3 (5^{2})$ (2) $5^{3}$ (3) $5^{4}$ (4) $2 (5^{2})$ 171. $The value of \frac{1 \times 2 ^{2} + 2 \times 3 ^{2} + \dots + 100 \times ( 101 ) ^{2}}{1 ^{2} \times 2 + 2 ^{2} \times 3 + \dots . + 100 ^{2} \times 101} is$ (1) $\frac{32}{31}$ (2) $\frac{31}{30}$ (3) $\frac{306}{305}$ (4) $\frac{305}{301}$ 172. $Let (a_{n}) be a sequence such that a_{0} = 0, a_{1} = \frac{1}{2} and 2 a_{n + 2} = 5 a_{n + 1} - 3 a_{n}, n 0, 1, 2, 3, \dots Then \sum_{k = 1}^{100} a_{k} is equal to$ (1) $3 a_{99} - 100$ (2) $3 a_{100} - 100$ (3) $3 a_{99} + 100$ (4) $3 a_{100} + 100$ 173. $Let a, b, c, d and p be any non zero distinct real numbers such that (a^{2} + b^{2} + c^{2}) p^{2} - 2 (ab + bc + cd) p + (b^{2} + c^{2} + d^{2}) = 0 then$ (1) $a, c, p are in G . P .$ (2) $a, b, c, d are in A . P .$ (3) $a, c, p are in A . P .$ (4) $a, b, c, d are in G . P .$ 174. $If the sum of first 11 terms of an A . P ., a_{1}, a_{2}, a_{3} \dots is 0 (a_{1} \neq = 0), then the sum of the A . P ., a_{1}, a_{3}, a_{5}, \dots, a_{23} is k a_{1} where k is equal to:$ (1) $- \frac{121}{10}$ (2) $- \frac{72}{5}$ (3) $\frac{72}{5}$ (4) $\frac{121}{10}$ 175. $For x ⩾ 0, the least value of K , for which 4^{1 + x} + 4^{1 - x}, \frac{K}{2}, 16^{x} + 16^{- x} are three consecutive terms of an A.P., is equal to:$ (1) $8$ (2) $4$ (3) $10$ (4) $16$ 176. $Let x_{1}, x_{2}, x_{3}, x_{4} be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x_{1}, x_{2}, x_{3}, x_{4}, then the resulting numbers are in an arithmetic progression. Then the value of \frac{1}{24} (x_{1} x_{2} x_{3} x_{4}) is:$ (1) $18$ (2) $216$ (3) $36$ (4) $72$ 177. $Let a_{1}, a_{2}, a_{3}, \dots be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a_{1} a_{2} + a_{2} a_{4} a_{9} + a_{3} + a_{7} is equal to __________$ 178. $If S (x) = (1 + x) + 2 (1 + x)^{2} + 3 (1 + x)^{3} + \dots + 60 (1 + x)^{60}, x \neq = 0, and (60)^{2} S (60) = a (b)^{b} + b, where a, b \in N, then (a + b) equal to______$ 179. $Let a_{1}, a_{2}, \dots ., a_{n} be in A.P. If a_{5} = 2 a_{7} and a_{11} = 18, then 12 (- \frac{1}{a _{10} + a _{11}} + \frac{1}{a _{11} + a _{12}} + \dots \frac{1}{a _{17} + a _{18}} is equal to _________$ 180. $Let a_{1}, a_{2}, a_{3}, \dots be in an arithmetic progression of positive terms. Let A_{k} = a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + \dots + a_{2 k - 1}^{2} - a_{2 k}^{2} . If A_{3} = - 153, A_{5} = - 435 and a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = 66, then a_{17} - A_{7} is equal to______$