Answer:
[tex]T_{10} - T_7 = 112[/tex]
Step-by-step explanation:
Given
[tex]T_9 = 63[/tex]
Required
Find [tex]T_{10} - T_7[/tex]
From the question, we have that:
Each sequence = 2 * Previous sequence + 1;
i.e.
[tex]T_n = 2 * T_{n - 1} + 1[/tex]
Considering the 9th sequence;
[tex]T_9 = 2 * T_8 + 1[/tex] ------ Equation 1
Considering the 8th sequence;
[tex]T_8 = 2 * T_7 + 1[/tex]
Substitute [tex]2 * T_7 + 1[/tex] for [tex]T_8[/tex] in equation 1
[tex]T_9 = 2 * T_8 + 1[/tex] becomes
[tex]T_9 = 2 * (2 * T_7 + 1) + 1[/tex]
Open bracket
[tex]T_9 = 2 * 2 * T_7 + 2*1 + 1[/tex]
[tex]T_9 = 4T_7 + 2 + 1[/tex]
[tex]T_9 = 4T_7 + 3[/tex]
Substitute 63 for [tex]T_9[/tex]
[tex]63 = 4T_7 + 3[/tex]
Subtract 3 from both sides
[tex]63 - 3 = 4T_3 + 3 - 3[/tex]
[tex]60 = 4T_3[/tex]
Divide both sides by 4
[tex]\frac{60}{4} = \frac{4T_3}{4}[/tex]
[tex]15 = T_7[/tex]
[tex]T_7 = 15[/tex]
Considering [tex]T_{10}[/tex]
[tex]T_1_0 = 2 * T_9 + 1[/tex]
Substitute 63 for [tex]T_9[/tex]
[tex]T_1_0 = 2 * 63 + 1[/tex]
[tex]T_1_0 = 126 + 1[/tex]
[tex]T_1_0 = 127[/tex]
Calculating [tex]T_{10} - T_7[/tex]
[tex]T_{10} - T_7 = 127 - 15[/tex]
[tex]T_{10} - T_7 = 112[/tex]
Hence, the 10th - 7th number is 112
