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Find the n^(th) terms of the series ...

Find the ` n^(th)` terms of the series
` 5 + 7 + 13 + 31 + …`

Text Solution

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Let ` S_(n) = 5 + 7 + 13 + 31 + …+ a_(n)` …(i)
Again , ` S_(n) = 5 + 7 + 13 + …+ a_(n-1) + a_(n)` …(ii)
On subtracting (ii) from (i), we have
` 0 = 5 + [ 2 + 6 + 18 + …+ (a_(n) - a_(n-1))]- a_(n)`
`rArr a_(n) = 5 + (2(3^(n-1) -1))/(3-1)`
`rArr a_(n) = 5 + 3^(n-1) - 1 `
` therefore a_(n) = 4 + 3 ^(n-1)`
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