Question

(iii) Find the sum of series 5 X 6 + 6 x7 + 7 X 8+ ... to 25 terms
use ap or gp​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Answer:

Step-by-step explanation:

The given series is (5 x 6) + (6 x 7) + (7 x 8) + . . . . . .  to 25 terms

Let the first term (5 x 6) be written as (n + 4) (n + 5)

=> The second term (6 x 7) = (n + 5) (n + 6)

We need to find the sum of 25 terms,

Therefore we need to find $\sum\limits_{n =1 }^{25}(n + 4) (n + 5)$

=>  $\sum\limits_{n =1 }^{25}(n^2 + 4n + 5n + 20)$

=>  $\sum\limits_{n=1 }^{25}(n^2 + 9n + 20)$

=>  $\sum\limits_{n=1 }^{25}n^2 + 9 \sum\limits_{n=1 }^{25}n + 20$

We know that $\sum n^2 = \frac{n(n+1)(2n+1)}{6}$ and $\sum n = \frac{n(n+1)}{2}$

=>   $\frac{25(25+1)(2*25+1)}{6} + \frac{25(25+1)}{2} + 20$

=> $\frac{25*26*51}{6} + \frac{25*26}{2} + 20$

=>  $\frac{25*26*51}{6} + \frac{25*26}{2} + 20$

=> 5525 + 325 + 20

=> 5870

Therefore, (5 x 6) + (6 x 7) + (7 x 8) + . . . . . .  to 25 terms = 5870