Sequence & Series PYQs Set 8 (PDF)

Set 8 of 10 · Questions 211-240 of 284

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Sequence & Series · Set 8 (Q211-240)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
211. $In an A.P., the sixth terms a_{6} = 2 . If the a_{1} a_{4} a_{5} is the greatest, then the common difference of the A.P., is equal to$ (1) $\frac{3}{2}$ (2) $\frac{8}{5}$ (3) $\frac{2}{3}$ (4) $\frac{5}{8}$ 212. $If \frac{1}{2. 3 ^{10}} + \frac{1}{2 ^{2} . 3 ^{9}} + ... \frac{1}{2 ^{10} .3} = \frac{K}{2 ^{10} . 3 ^{10}}, then the remainder when K is divided by 6 is$ (1) $1$ (2) $2$ (3) $3$ (4) $5$ 213. $Let S_{n} be the sum of the first n terms of an arithmetic progression. If S_{3 n} = 3 S_{2 n}, then the value of \frac{S _{4 n}}{S _{2 n}} is :$ (1) $6$ (2) $4$ (3) $2$ (4) $8$ 214. $Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then :$ (1) $P^{2} = 63 Q$ (2) $P^{2} = 363 Q$ (3) $P = 363 Q^{2}$ (4) $P^{2} = 723 Q$ 215. $If x = \sum_{n = 0}^{\infty} a^{n}, y = \sum_{n = 0}^{\infty} b^{n}, z = \sum_{n = 0}^{\infty} c^{n}, where a, b, c are in A.P. and \|a\| < 1, \|b\| < 1, \|c\| < 1, abc \neq = 0, then$ (1) $x, y, z are in A.P.$ (2) $x, y, z are in G.P.$ (3) $\frac{1}{x}, \frac{1}{y}, \frac{1}{z} are in A.P.$ (4) $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 - (a + b + c)$ 216. $Let s_{1}, s_{2}, s_{3}, \dots., s_{10} respectively be the sum of 12 terms of 10 A. Ps whose first terms are 1, 2, 3,\dots.,10 and the common differences are 1, 3, 5,..., 19 respectively. Then \sum_{i = 1}^{10} S_{i} equal to$ (1) $7220$ (2) $7360$ (3) $7260$ (4) $7380$ 217. $Let S = 2 + \frac{6}{7} + \frac{12}{7 ^{2}} + \frac{20}{7 ^{3}} + \frac{30}{7 ^{4}} + ... then 4S is equal to$ (1) $(\frac{7}{3})^{2}$ (2) $\frac{7 ^{3}}{3 ^{2}}$ (3) $(\frac{7}{3})^{3}$ (4) $\frac{7 ^{3}}{3 ^{3}}$ 218. $If (10)9 + 2(11)1 (10)8 + 3(11)2 (10)7 + . . . . . . . . + 10 (11)9 = k(10)9, then k is equal to$ (1) $100$ (2) $110$ (3) $\frac{121}{10}$ (4) $\frac{441}{100}$ 219. $The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41+... is$ (1) $3520$ (2) $3450$ (3) $3250$ (4) $3420$ 220. $Let a_{1}, a_{2}, a_{3},\dots\dots, a_{49} be in A.P. such that \sum_{k = 0}^{12} a_{4 k + 1} = 416 and a_{9} + a_{43} = 66 . If a_{1}^{2} + a_{2}^{2} + ...... + a_{1}^{2}_{7} = 140 m, then m is equal to$ (1) $34$ (2) $33$ (3) $66$ (4) $68$ 221. $If sum of the first 21 terms of the series, lo g_{9^{1/2}} x + lo g_{9^{1/3}} x + lo g_{9^{1/4}} x + ... where x>0 is 504, then x is equal to$ (1) $243$ (2) $9$ (3) $7$ (4) $81$ 222. $Let bi > 1 for i = 1,2,\dots.,101. Suppose logeb1,logeb2,\dots,logeb101 are in Arithmetic progression (A.P.) with the common difference loge 2. Suppose a1, a2,\dots,a101 are in A.P. such that a1 = b1 and a51= b51. If t = b1 + b2 + \dots. + b51 and s = a1 + a2 + \dots + a51, then$ (1) $s > t and a101 > b101$ (2) $s > t and a101 < b101$ (3) $s < t and a101 > b101$ (4) $s < t and a101 < b101$ 223. $Suppose a,, a_{2}, ...., a_{n},... be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is 5 : 17 and 110 < a_{15} < 120, then the sum of the first ten terms of the progression is equal to -$ (1) $290$ (2) $380$ (3) $460$ (4) $510$ 224. $Let a,b,c \in R. If f(x) = a x^{2} + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, \forall x,y \in R, then \sum_{n = 1}^{10} f (n) is equal to$ (1) $330$ (2) $165$ (3) $190$ (4) $225$ 225. $Let {a_{n}} be a sequence such that a_{0} = a_{1} = 0 and a_{n + 2} = 3 a_{n + 1} - 2 a_{n} + 1, \forall n \geq 0 . Then a_{25} a_{23} - 2 a_{25} a_{22} - 2 a_{23} a_{24} + 4 a_{22} a_{24} is equal to:$ (1) $483$ (2) $528$ (3) $575$ (4) $624$ 226. $The least value of α\inR for which 4αx2 + \frac{1}{x} \geq 1, for all x > 0, is$ (1) $\frac{1}{64}$ (2) $\frac{1}{32}$ (3) $\frac{1}{27}$ (4) $\frac{1}{25}$ 227. $Let S_{K} = \frac{1 + 2 + ... + K}{K} and where A, B, C, D \in N and A Has least value then$ (1) $A + C + D is not divisible by D$ (2) $A + B = 5(D - C)$ (3) $A + B + C + D is divisible by 5$ (4) $A + B is divisible by D$ 228. $The sum 1 + 2 . 3 +3 . 3^{2} +.....+ 10 . 3^{9} is equal to$ (1) $\frac{2. 3 ^{12} + 10}{4}$ (2) $\frac{19. 3 ^{10} + 1}{4}$ (3) $5. 3^{10} -2$ (4) $\frac{9. 3 ^{10} + 1}{2}$ 229. $Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is$ 230. $Let the positive integers be written in the form : If the k^{th} row contains exactly k numbers for every natural number k, then the row in which the number 5310 will be, is_____$ 231. $If the sum of the first ten terms of the series \frac{1}{5} + \frac{2}{65} + \frac{3}{325} + \frac{4}{1025} + \frac{5}{2501} + ... is \frac{m}{n}, where m and n are co-prime numbers, then m + n is equal$ 232. $The sum of all those terms, of the arithmetic progression 3, 8, 13, ..., 373, which are not divisible by 3, is equal to________.$ 233. $Let S = 109 + \frac{108}{5} + \frac{107}{5 ^{2}} + .... + \frac{2}{5 ^{107}} + \frac{5}{5 ^{108}} Then the value of (16S - (25)^{- 54}) is equal to$ 234. $For k \in N, if the sum of the series 1 + \frac{4}{k} + \frac{8}{k ^{2}} + \frac{13}{k ^{3}} + \frac{19}{k ^{4}} + ... is 10, then the value of k is$ 235. $Let a_{1} = b_{1} = 1, a_{n - 1} + 2 and b_{n} - a_{n} + b_{n - 1} : for every natural number n \geq 2. Then \sum_{n = 1}^{15} a_{n} . b_{n} is equal to __________.$ 236. $The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?$ 237. $The minimum value of the sum of real numbers a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8} and a^{10} where a > 0 is$ 238. $Let a_{1,} a_{2}, a_{3},... be an A.P. If \sum_{r = 1}^{\infty} \frac{a _{r}}{2 ^{r}} = 4, then 4 a_{2} is equal to________.$ 239. $If the sum of the series (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{2 ^{2}} - \frac{1}{2.3} + \frac{1}{3 ^{2}}) + (\frac{1}{2 ^{3}} - \frac{1}{2 ^{2} .3} + \frac{1}{2. 3 ^{2}} - \frac{1}{3 ^{3}}) + (\frac{1}{2 ^{4}} - \frac{1}{2 ^{3} .3} + \frac{1}{2 ^{2} . 3 ^{2}} - \frac{1}{2. 3 ^{3}} + \frac{1}{3 ^{4}}) + ... is \frac{α}{β},where α and β are co-prime, then α + 3 β is equal to _____$ 240. $An arithmetic progression is written in the following way The sum of all the terms of the 10^{th} row is______$