Question

The first difference of a sequence is 5, 8, 11, 14,... The sum of the first two terms of the original sequence is 23. Find the first three terms of the original sequence

Answer 1

The first three terms of the original sequence are 9, 14, and 22.

Explanation:

From the question,

The first difference of the sequence is 5, 8, 11, 14,... That is,

`T_(2) - T_(1) = 5\\T_(3) - T_(2) = 8\\T_(4) - T_(3) = 11\\T_(5) - T_(4) = 14`

Where
`T_(1)` is the first term of the original sequence

`T_(2)` is the second term of the original sequence

`T_(3)` is the third term of the original sequence etc.

Also, from the question, the sum of the first two terms of the original sequence is 23; that is,

`T_(1) + T_(2) = 23`

Now, we can find
`T_(1)` and
`T_(2)` by solving the following equations simultaneously

`T_(2) - T_(1) = 5` .......... (1)

`T_(1) + T_(2) = 23` ......... (2)

From equation (1)

`T_(2) - T_(1) = 5`

Then,

`T_(2) = 5 + T_(1)` .......... (3)

Substitute the value of
`T_(2)` into equation (2)

Then,
`T_(1) + T_(2) = 23` becomes

`T_(1) + 5 + T_(1) = 23\\`

Then,

`2T_(1) + 5 = 23\\2T_(1) = 23 - 5\\2T_(1) = 18`

`T_(1) = (18)/(2) \\T_(1) = 9`

Hence, the first term of the original sequence is 9

Now, substitute the value of
`T_(1)` into equation (3)

Then,
`T_(2) = 5 + T_(1)` become

`T_(2) = 5 + 9\\`

∴
`T_(2) = 14`

Hence, the second term of the original sequence is 14

The third term of the original sequence is given by

`T_(3) - T_(2) = 8`

Then,

`T_(3) = 8 + T_(2)`

`T_(3) = 8 + 14\\T_(3) = 22`

Hence, the third term of the original sequence is 22.

Hence, the first three terms of the original sequence are 9, 14, and 22.