Question

sum of the series 3 + 5 + 7 +
+ (2n + 1) ....​

Answer 1

There are n terms in the series . Because last odd term =2n-1 and rth odd term is denoted by 2r-1 .

We can also calculate number of terms by formula.

Let there are r terms in the series .

ar = a + (r-1)d

ar = rth term =2n-1

a=first term =1

d=common difference = 2

2n-1 =1+ 2(r-1)

2n-1=1+2r -2

2n=2r

r=n

so number of terms =n

The sum of n terms of series

=n/2 [2a + (n-1)d]

=n/2 [2(1) +2(n-1)]

=n/2 [2n]

=n²