Question

what is the sum of the series 1cube-2cube + 3cube-4 cube +.....9 cube.​

Answer 1

We have to find the sum of series ..

1³ - 2³ + 3³ - 4³ + .... 9³

Solution : here series is ...

1³ - 2³ + 3³ - 4³ + .... + 9³

= 1³ - 2³ + 3³ - 4³ + ..... -8³ + 9³

= (1³ + 3³ + ... + 9³) - (2³ + 4³ + ... 8³)

= 1 + (3³ + 5³ + .. + 9³) - (2³ + 4³ + ... + 8³)

= 1 + $\Sigma(2r+1)^3-\Sigma(2r)^3$ , where r is integers 1 to 4.

= 1 + $\Sigma\{(2r+1)^3-(2r)^3\}$

= 1 + $\Sigma\{12r^2+6r+1\}$

= 1 + $12\Sigma r^2+6\Sigma r+\Sigma1$

= 1 + 12n(n + 1)(2n + 1)/6 + 6n(n + 1)/2 + n

= 1 + 2n(n + 1)(2n + 1) + 3n(n + 1) + n

Now put n = 4

= 1 + 2 × 4 × 5 × 9 + 3 × 4 × 5 + 4

= 1 + 360 + 60 + 4

= 425

Therefore the sum of given series is 425.