Question

Find the sum up to the 17^th term of the series `1^3/1 + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ...` 

Answer 1

t₁ = `1^3/1`

t₃ = `(1^3 + 2^3)/(1 + 3)`

t₃ = `(1^3 + 2^3 + 3^3)/(1 + 3 + 5)`

∴ t_n = `(1^3 + 2^3 + 3^3 + ... :n:^3)/(1 + 3 + 5 + ... "n terms")`

= `(sum"n"^3)/("n"^2) ((2"n" - 1 + 1)/2)^2`

= `("n"^2("n" + 1)^2)/(4("n"^2))`

= `("n" + 1)^2/4`

= `("n"^2 + 2"n" + 1)/4`

∴ S_n = `sum"t"_"n" = 1/4 sum"n"^2 + 2"n" + 1`

= `1/4 sum"n"^2 + sum2"n" + sum1`

= `1/4[("n"("n" + 1)(2"n" + 1))/6 + (2("n")("n" + 1))/2 + "n"]`

To find S₁₇ put n = 17

S₁₇ = `1/4[(17 xx 18 xx 35)/6 + (2(17)(18))/2 + 17]`

= `1/4[17 xx 105 + 17 xx 18 + 17]`

= `(17(105 + 18 + 1))/4`

= 17 × 31

= 527