Question

Each number in a sequence is formed by doubling the previous number and then adding 1. If the ninth number in the sequence is63, what is the 10th number minus the 7th number?

Answer 1

`T_(10) - T_7 = 112`

Explanation:

Given

`T_9 = 63`

Required

Find
`T_(10) - T_7`

From the question, we have that:

Each sequence = 2 * Previous sequence + 1;

i.e.

`T_n = 2 * T_(n - 1) + 1`

Considering the 9th sequence;

`T_9 = 2 * T_8 + 1` ------ Equation 1

Considering the 8th sequence;

`T_8 = 2 * T_7 + 1`

Substitute
`2 * T_7 + 1` for
`T_8` in equation 1

`T_9 = 2 * T_8 + 1` becomes

`T_9 = 2 * (2 * T_7 + 1) + 1`

Open bracket

`T_9 = 2 * 2 * T_7 + 2*1 + 1`

`T_9 = 4T_7 + 2 + 1`

`T_9 = 4T_7 + 3`

Substitute 63 for
`T_9`

`63 = 4T_7 + 3`

Subtract 3 from both sides

`63 - 3 = 4T_3 + 3 - 3`

`60 = 4T_3`

Divide both sides by 4

`(60)/(4) = (4T_3)/(4)`

`15 = T_7`

`T_7 = 15`

Considering
`T_(10)`

`T_1_0 = 2 * T_9 + 1`

Substitute 63 for
`T_9`

`T_1_0 = 2 * 63 + 1`

`T_1_0 = 126 + 1`

`T_1_0 = 127`

Calculating
`T_(10) - T_7`

`T_(10) - T_7 = 127 - 15`

`T_(10) - T_7 = 112`

Hence, the 10th - 7th number is 112