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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?

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Answer:


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

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