Find the sum of $20$ terms of a series of which every even term is $2$ times the term the term before it and every odd terms is $3$ times the before it and the first term is unity ?

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Solution

A/Q the series is in the form of
$1, 2, 6, 12, 36, . . . . . . . . .20 t e r m s ∴ S_{20} = (1 + 2 + 6 + 12 + 36 + . . . . . . . + u p t o 20 t e r m s) \Rightarrow S_{20} = (1 + 6 + 36 + . . . . . + u p t o 10 t e r m s) + (2 + 12 + 72 + . . . . . . . + u p t o 10 t e r m s)$
both the sequence is in $G P$
$∴ S_{20} = \frac{a (r^{10} - 1)}{r - 1} + \frac{a (r^{10} - 1)}{r - 1} \Rightarrow S_{20} = \frac{1 (6^{10} - 1)}{6 - 1} + \frac{1 (6^{10} - 1)}{6 - 1} \Rightarrow S_{20} = 36279705$

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

A/Q the series is in the form of 1,2,6,12,36,.........20terms∴S20=(1+2+6+12+36+.......+upto20terms)⇒S20=(1+6+36+.....+upto10terms)+(2+12+72+.......+upto10terms)both the sequence is in GP∴S20=a(r10−1)r−1+a(r10−1)r−1⇒S20=1(610−1)6−1+1(610−1)6−1⇒S20=36279705

Find the sum of $20$ terms of a series of which every even term is $2$ times the term the term before it and every odd terms is $3$ times the before it and the first term is unity ?