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Let a1, a2, a3,... be an arithmetic progression with first term a1 = 7 and common difference 8. Define a sequence T1, T2, T3,... such that T1 = 3 and T(n+1) - T(n) = a(n) for all n >= 1. Which of the following statements are TRUE?

1. T20 = 1504
2. T(n+1) = 4*n² + 2*n + 3 for n >= 0
3. T30 = 3454
4. T(n) = 4*n² - 5*n + 4 for n >= 1

Correct answer: T20 = 1504

Solution

Using telescoping, T(n) = 3 + sumₖ₌₁ⁿ⁻¹(8k-1) = 3 + 4(n-1)n - (n-1) = 4n² - 5n + 4, giving T20 = 1444, T30 = 3454.

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