Question

The arithmetic sequence a_ia

i

​

a, start subscript, i, end subscript is defined by the formula:

a_1 = 5a

1

​

=5a, start subscript, 1, end subscript, equals, 5

a_i = a_{i - 1} + 2a

i

​

=a

i−1

​

+2a, start subscript, i, end subscript, equals, a, start subscript, i, minus, 1, end subscript, plus, 2

Find the sum of the first 700700700 terms in the sequence.

Answer 1

492,800

Explanation:

Given ith term of an arithmetic sequence as shown:

ai = a(i-1)+2

and a1 = 5

When i = 2

a2 = a(2-1)+2

a2 = a1+2

a2 = 5+2

a2 = 7

When i = 3

a3 = a(3-1)+2

a3 = a2+2

a3 = 7+2

a3 = 9

It can be seen that a1, a2 and a3 forms an arithmetic progression

5,7,9...

Given first term a1 = 5

Common difference d = 7-5= 9-7 = 2

To calculate the sum of the first 700 of the sequence, we will use the formula for finding the sum of an arithmetic sequence.

Sn = n/2{2a1+(n-1)d}

Given n = 700

S700 = 700/2{2(5)+(700-1)2}

S700 = 350{10+699(2)}

S700 = 350{10+1398}

S700 = 350×1408

S700 = 492,800

Therefore, the sum of the first 700 terms in the sequence is 492,800