Question

The sum of n term of the series 2.3^2+5.4^2+8.5^2+............. is​

Answer 1

The sum of n term of the series 2.3² + 5.4² + 8.5² + ....... is

solution : This series is the combination of two different series. 2.3² + 5.4² + 8.5² + .......

Have a look bold underlined number of given series, i.e., 2 , 5 , 8 ..... this is an arithmetic progression.

find r th term of this ap.

Tr = 2 + (r - 1)3 = 2 + 3r - 3 = (3r - 1) .....(1)

now think about 3² , 4² , 5² ....

r th term must be (r + 2)²

Then 2.3² + 5.4² + 8.5² + ....... = $\Sigma(3r-1)(r+2)$

= $\Sigma (3r^2+6r-r-2)$

= $\Sigma(3r^2+5r-2)$

= $3\Sigma r^2+5\Sigma r-2\Sigma$

= 3[n(n + 1)/2]² + 5[n(n + 1)/2] - 2n

= 3n²(n + 1)²/4 + 5n(n + 1)/2 - 2n

= n/4 [3n³ + 6n³ + 13n + 2]

Therefore the sum of the series is n/4 [3n³ + 6n³ + 13n + 2] .