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Maths-

General

Easy

Question

The number of terms common between the series $1 plus 2 plus 4 plus 8 plus horizontal ellipsis$ to 100 terms and $1 plus 4 plus 7 plus 10 plus horizontal ellipsis$ to 100 terms is

6
4
5
None of these

The correct answer is: 5

For G.P., $t subscript n end subscript equals 2 to the power of n minus 1 end exponent$ ; for A.P. $T subscript m end subscript equals 1 plus open parentheses m minus 1 close parentheses 3 equals 3 m minus 2$
They are common if $2 to the power of n minus 1 end exponent equals 3 m minus 2$ . For G.P. 100^th term is $2 to the power of 99 end exponent$ . For A.P. 100^th term is $1 plus open parentheses 100 minus 1 close parentheses 3 equals 298$ . Now we must choose value of $m$ such that $3 m minus 2$ is of type $2 to the power of n minus 1 end exponent$ . Hence, $3 m minus 2 equals 1 comma blank 2 comma blank 4 comma blank 8 comma blank 16 comma blank 32 comma blank 64 comma blank 128 comma blank 256$ for which $m equals 1 comma blank$ 4/3, 2, 10/3, 6, 34/2, 22, 130/3, 86. Hence, possible values of $m$ are 1, 2, 6, 22, 86. Hence, there are five common terms

Related Questions to study

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