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Question

The sum of 20 terms of a series of which every even term is 2 times the term before it, and every odd term is 3 times the term before it, the first term being unity is

$(\frac{2}{7}) (6^{10} - 1)$
$(\frac{3}{7}) (6^{10} - 1)$
$(\frac{3}{5}) (6^{10} - 1)$
None of these

The correct answer is: $open parentheses fraction numerator 3 over denominator 5 end fraction close parentheses left parenthesis 6 to the power of 10 end exponent minus 1 right parenthesis$

The series is
$1 plus 2 plus 2 cross times 3 plus 2 to the power of 2 end exponent cross times 3 plus 2 to the power of 2 end exponent cross times 3 to the power of 2 end exponent plus 2 to the power of 3 end exponent cross times 3 to the power of 2 end exponent plus horizontal ellipsis$ to 20 terms
$equals left parenthesis 1 plus 2 cross times 3 plus 2 to the power of 2 end exponent cross times 3 to the power of 2 end exponent plus horizontal ellipsis$ to 10 terms) $plus left parenthesis 2 plus 2 to the power of 2 end exponent cross times 3 plus 2 to the power of 3 end exponent cross times 3 to the power of 2 end exponent plus horizontal ellipsis$ to 10 terms)
$equals fraction numerator 1 left parenthesis 2 to the power of 10 end exponent 3 to the power of 10 end exponent minus 1 right parenthesis over denominator 6 minus 1 end fraction plus fraction numerator 2 left parenthesis 2 to the power of 10 end exponent 3 to the power of 10 end exponent minus 1 right parenthesis over denominator 6 minus 1 end fraction$
$equals open parentheses fraction numerator 3 over denominator 5 end fraction close parentheses left parenthesis 6 to the power of 10 end exponent minus 1 right parenthesis$

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