Question

Find the sum of the arithmetic sequence. 3,5,7,9,....,21

Answer 1

The sum of any arithmetic sequence is the average of the first and last terms times the number of terms.

Any term in an arithmetic sequence is:

a(n)=a+d(n-1), where a=initial term, d=common difference, n=term number

So the first term is a, and the last term is a+d(n-1) so the sum can be expressed as:

s(n)(a+a+d(n-1))(n/2)

s(n)=(2a+dn-d)(n/2)

s(n)=(2an+dn^2-dn)/2

However we need to know how many terms are in the sequence.

a(n)=a+d(n-1), and we know a=3 and d=2 and a(n)=21 so

21=3+2(n-1)

18=2(n-1)

9=n-1

10=n so there are 10 terms in the sequence.

s(n)=(2an+dn^2-dn)/2, becomes, a=3, d=2, n=10

s(10)=(2*3*10+2*10^2-2*10)/2

s(10)=(60+200-20)/2

s(10)=240/2

s(10)=120