1 Arithmetic sequence / 3 pts
An arithmetic sequence satisfies \(a_3 = 11\) and \(a_8 = 31\). (a) Find the common difference. (b) Find \(a_1\) and a formula for \(a_n\). (c) Find \(S_{25}\).
2 Geometric sequence / 3 pts
A geometric sequence has \(a_2 = 12\) and \(a_5 = 324\). Find the ratio \(r\), the first term \(a_1\), and the sum of the first 7 terms.
3 Infinite series / 3 pts
(a) Find the sum of \(20 - 10 + 5 - \dfrac52 + \cdots\). (b) Write \(0.272727\ldots\) as a fraction in lowest terms.
4 Sigma notation and recursion / 3 pts
(a) Expand and evaluate \(\displaystyle\sum_{k=1}^{6} 2^k\). (b) A sequence has \(b_1 = 2\) and \(b_n = b_{n-1} + 2n\) for \(n \ge 2\). Compute \(b_2, b_3, b_4\) and conjecture a formula for \(b_n\).
5 Proof by induction / 4 pts
Prove by induction that \(1 + 3 + 9 + \cdots + 3^{n-1} = \dfrac{3^n - 1}{2}\) for every integer \(n \ge 1\).
6 Binomial Theorem / 4 pts
(a) Expand \((x - 2)^5\). (b) Find the coefficient of \(x^3\) in \((1 + 2x)^7\).
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