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Question

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ be terms of an A.P. If $fraction numerator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript p end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis plus a subscript q end subscript end fraction equals fraction numerator p to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction comma blank p not equal to q$ , then $fraction numerator a subscript 6 end subscript over denominator a subscript 21 end subscript end fraction$ equals

41/11
7/2
2/7
11/41

The correct answer is: 11/41

Given, $a subscript 1 end subscript comma a subscript 2 end subscript comma a subscript 3 end subscript comma horizontal ellipsis$ are terms of A.P.
$therefore blank fraction numerator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis a subscript p end subscript over denominator a subscript 1 end subscript plus a subscript 2 end subscript plus horizontal ellipsis a subscript q end subscript end fraction equals fraction numerator p to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction$
$rightwards double arrow fraction numerator fraction numerator p over denominator 2 end fraction left square bracket 2 a subscript 1 end subscript plus open parentheses p minus 1 close parentheses d right square bracket over denominator fraction numerator q over denominator 2 end fraction left square bracket 2 a subscript 1 end subscript plus open parentheses q minus 1 close parentheses d right square bracket end fraction equals fraction numerator p to the power of 2 end exponent over denominator q to the power of 2 end exponent end fraction$
$rightwards double arrow fraction numerator 2 a subscript 1 end subscript plus open parentheses p minus 1 close parentheses d over denominator 2 a subscript 1 end subscript plus open parentheses q minus 1 close parentheses d end fraction equals fraction numerator p over denominator q end fraction$
$rightwards double arrow open square brackets 2 a subscript 1 end subscript plus open parentheses p minus 1 close parentheses d close square brackets q equals left square bracket 2 a subscript 1 end subscript plus open parentheses q minus 1 close parentheses d right square bracket p$
$rightwards double arrow 2 a subscript 1 end subscript open parentheses q minus p close parentheses equals d left square bracket open parentheses q minus 1 close parentheses p minus open parentheses p minus 1 close parentheses q right square bracket$
$rightwards double arrow 2 a subscript 1 end subscript open parentheses q minus p close parentheses equals d left square bracket open parentheses q minus p close parentheses$
$rightwards double arrow 2 a subscript 1 end subscript equals d$
$therefore fraction numerator a subscript 6 end subscript over denominator a subscript 21 end subscript end fraction equals fraction numerator a subscript 1 end subscript plus 5 d over denominator a subscript 1 end subscript plus 20 d end fraction equals fraction numerator a subscript 1 end subscript plus 10 a subscript 1 end subscript over denominator a subscript 1 end subscript plus 40 a subscript 1 end subscript end fraction equals fraction numerator 11 over denominator 41 end fraction$

Related Questions to study

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Consider the ten numbers $a r comma blank a r to the power of 2 end exponent comma a r to the power of 3 end exponent comma horizontal ellipsis comma a r to the power of 10 end exponent$ . If their sum is 18 and the sum of their reciprocals is 6 then the product of these ten numbers, is

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parallel

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parallel

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