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A sequence of number a1, a2, a3,…..,an i...

A sequence of number `a_1, a_2, a_3,…..,a_n` is generated by the rule `a_(n+1) = 2a_(n)`. If `a_(7) - a_(6) = 96`, then what is the value of `a_(7)` ?

A

48

B

96

C

98

D

192

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will use the given information about the sequence defined by the rule \( a_{n+1} = 2a_n \) and the condition \( a_7 - a_6 = 96 \). ### Step 1: Understand the recursive relationship The sequence is defined such that each term is twice the previous term. This means: - \( a_2 = 2a_1 \) - \( a_3 = 2a_2 = 2(2a_1) = 4a_1 \) - \( a_4 = 2a_3 = 2(4a_1) = 8a_1 \) - \( a_5 = 2a_4 = 2(8a_1) = 16a_1 \) - \( a_6 = 2a_5 = 2(16a_1) = 32a_1 \) - \( a_7 = 2a_6 = 2(32a_1) = 64a_1 \) ### Step 2: Set up the equation using the given condition We know from the problem that: \[ a_7 - a_6 = 96 \] Substituting the expressions we found: \[ 64a_1 - 32a_1 = 96 \] ### Step 3: Simplify the equation Now, simplify the left side: \[ (64a_1 - 32a_1) = 32a_1 \] So the equation becomes: \[ 32a_1 = 96 \] ### Step 4: Solve for \( a_1 \) To find \( a_1 \), divide both sides by 32: \[ a_1 = \frac{96}{32} = 3 \] ### Step 5: Calculate \( a_7 \) Now that we have \( a_1 \), we can find \( a_7 \): \[ a_7 = 64a_1 = 64 \times 3 = 192 \] ### Conclusion Thus, the value of \( a_7 \) is \( \boxed{192} \). ---
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