Find the sum to n terms of each of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + …

Asked by Abhisek | 1 year ago |  92

##### Solution :-

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)

= n+ 3n+ 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 ago

### Related Questions

#### Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

#### Find the two numbers whose A.M. is 25 and GM is 20.

Find the two numbers whose A.M. is 25 and GM is 20.

#### If a is the G.M. of 2 and 1/4 find a.

If a is the G.M. of 2 and $$\dfrac{1}{4}$$ find a.

#### Find the geometric means of the following pairs of numbers

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 $$\dfrac{81}{2}$$.