sandy wrote:
In a certain sequence, the term an is defined by the formula \(a_n = 2 \times a_{n - 1}\) for each integer n ≥ 2. If \(a_1 = 1\), what is the positive difference between the sum of the first 10 terms of the sequence and the sum of the 11th and 12th terms of the same sequence?
(A) 1
(B) 1,024
(C) 1,025
(D) 2,048
(E) 2,049
Each term after the first is twice the preceding term.
First 10 terms:
1,
2, 4, 8, 16, 32, 64, 128, 256, 512Since the first term is ODD and remaining terms are all EVEN, the sum of the first 10 terms = ODD + EVEN = ODD
11th and 12th terms:
1024, 2048
Sum = EVEN + EVEN = EVEN
Difference between the second sum and the first sum = EVEN - ODD = ODD
Since the correct answer must be ODD, eliminate B and D.
Rather than calculate, BALLPARK the difference between the two sums:
11th term + 12th term ≈ 1000 + 2000 = 3000
Sum of the first 10 terms ≈ 500 + 250 + 125 + (more than 100) ≈ 1000
Thus:
Approximate difference between the sums = 3000 - 1000 = 2000
Of the remaining answer choices, only E is viable.
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