Question

The first difference of a sequence is the arithmetic sequence 1​, 3​, 5​, 7​, 9​, .... Find the first six terms of the original sequence in each of the following cases. a. The first term of the original sequence is 1. b. The sum of the first two terms in the original sequence is 5. c. The fifth term in the original sequence is 28.

Answer 1

Explanation:

Given that the first difference of a sequence is the arithmetic sequence 1​, 3​, 5​, 7​, 9​, ....

a) When I term a =1

`a_2 =1+1 =2\\a_3 = 4+5 =9\\a_4 = 9+7 =16\\a_5 =16+9 =25\\a_6=25+11 =36`

Thus first 6 terms are

1,2,5,12,21,32.....

b) Here
`a_1+a_2=5\\a_2-a_1 =3\\-------------\\2a_2=8\\a_2 =4\\a_1 =1`

`a_2 =1+3 =4\\a_3 = 4+5 =9\\a_4 = 9+7 =16\\a_5 =16+9 =25\\a_6=25+11 =36`

So sequence would be

3,4,9,16,25, 36,...

c) When 5th term is 28

we have the sequences as

a1, a1+1,a1+1+3, ...a1+1+3+5+7

When 5th term is 28 we have

`a_1 +16 =28\\a_1 =12\\`

Hence first 6 terms would be

12, 13, 16, 21, 28, 37,...