Q. 53.8( 4 Votes )

# Sum the following series to n terms :

1 + 3 + 6 + 10 + 15 + …………….

Answer :

Let s = 1 + 3 + 6 + 10 + 15 + …………. + n

By shifting each term by one

S = 1 + 3 + 6 + 10 + 15 + …………. + nth …………… (1)

S = 1 + 3 + 6 + 10 + …………. + (n - 1)th + nth ….(2)

by (1) - (2) we get

0 = 1 + (2 + 3 + 4 + 5 + …….nth - (n - 1)th - n)

Nth = 1 + (2 + 3 + 4 + 5 + …….nth - (n - 1)th - nth)

Nth = 1 + (2 + 3 + 4 + …….r + 1) ……………(3)

Nth = 1 + (summation upto (n - 1)th term)

we know

Substituting the above-given value in (3)

thus

S =3 + 5 + 9 + 15 + 23 + …………. + nth =

We know by property that:

∑ax^{n} + bx^{n - 1} + cx^{n - 2}…….d_{0}=a∑x^{n} + b∑x^{n - 1} + c∑x^{n - 2}…….. + d_{0}∑1

…… (4)

We know

Thus substituting the above values in(4)

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