Given series is 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + …
It’s seen that,
nth term, an = n ( n + 1) ( n + 2)
= (n2 + n) (n + 2)
= n3 + 3n2 + 2n
Then, the sum of n terms of the series can be expressed as
\(s_n= \displaystyle\sum_{k=1}^{n} a_k\)
The sum of n terms of the given series is
\( s_n= \displaystyle\sum_{k=1}^{n} k(k+1)\)
\(\displaystyle\sum_{k=1}^{n} k^2+s_n= \displaystyle\sum_{k=1}^{n} k\)
\( \dfrac{n(n+1)(2n+1)}{6}+\dfrac{n(n+1)}{2}\)
= \( \dfrac{n(n+1)}{2}(\dfrac{2n+1}{3}+1)\)
= \( \dfrac{n(n+1)}{2}\dfrac{2n+4}{3}\)
= \( \dfrac{n(n+1)(n+2)}{3}\)
Answered by Abhisek | 1 year agoConstruct a quadratic in x such that A.M. of its roots is A and G.M. is G.
Find the geometric means of the following pairs of numbers:
(i) 2 and 8
(ii) a3b and ab3
(iii) –8 and –2
Insert 5 geometric means between \( \dfrac{32}{9}\) and \( \dfrac{81}{2}\).