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

Answer:

$T_{10} - T_7 = 112$

Step-by-step 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

$\frac{60}{4} = \frac{4T_3}{4}$

$15 = T_7$

$T_7 = 15$

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