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प्रश्न
Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 42) + (1 + 4 + 42 + 43) + ...
उत्तर
T1 = 1
T2 = 1 + 4
T3 = 1 + 4 + 42
Tn = 1 + 4 + 42 + ... n terms which is a G.P.
∴ Tn = `(1(4^"n" - 1))/(4 - 1)`
= `(4^"n" - 1)/3`
So, Sn = `sum"t"_"n"`
= `(sum4^"n" - 1)/(sum3)`
= `(sum4^"n" - sum1)/(sum3)`
`sum4^"n" = 4^1 + 4^2 + ....... "n terms"`
= `(4(4^"n" - 1))/(4 - 1)`
= `(4(4^"n" - 1))/3`
So `(sum4^"n" - sum1)/(sum3)`
= `((4(4^"n" - 1)- "n")/3)/3`
= `(4(4^"n" - 1) - 3"n")/9`
= `4/9(4^"n" - 1) - "n"/3`
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