Question

Find the sum of first 10 terms for the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …….​

Answer 1

Answer:

so series is 2,6,12,20

so it a geometric progression

2+6+12+20+...

245

Answer 2

The sum of first 10 terms for the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …….​ is 440.

Here, according to the given information, we are given that,

The series is given as,

1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …….​

Now, the series can be written as,

1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …….​

= 2 + 6 + 12 + 20 + ....

Then, the nth term is n(n+1) since all the terms of the given series are of the form n(n+1).

This means that the nth term is of the form $n(n+1) = n^{2} +n$.

Now, if we take summation, we get,

∑$(n^{2} +n)$ = ∑$n^{2}$  + ∑$n$

This gives us,

∑$(n^{2} +n)$ = $\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}$

= $\frac{n(n+1)}{2} .\frac{2n+4}{3}$

=$\frac{n(n+1)(n+2)}{3}$

Now, taking n equal to 10, we get,

$\frac{n(n+1)(n+2)}{3}$

$=\frac{10(10+1)(10+2)}{3}\\=\frac{10.11.12}{3} \\=440$

Hence, the sum of first 10 terms for the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …….​ is 440.

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