Sequences and Series: math test solutions, Grade 11 – download the PDF

1 Two sequences / 4 pts

(a) Arithmetic, \(d = 5\): \(a_n = 11 + 5(n-1) = 5n + 6\) (1 pt). \(a_{25} = 131\) (1 pt).

(b) Geometric, \(r = \dfrac13\): \(a_n = 243\left(\dfrac13\right)^{n-1}\) (1 pt). \(a_6 = 243\cdot\dfrac{1}{243} = 1\) (1 pt).

2 Recursive rule / 3 pts

(a) \(4, 1, -2, -5\) (1 pt).

(b) \(a_n = 4 - 3(n-1) = 7 - 3n\) (1 pt).

(c) \(a_{30} = 7 - 90 = -83\) (1 pt).

3 Arithmetic series / 4 pts

(a) \(d = 4\), \(a_{30} = 9 + 29\cdot 4 = 125\) (1 pt). \(S_{30} = \dfrac{30\,(9 + 125)}{2} = 2{,}010\) (1 pt).

(b) \(S_n = \dfrac{n[18 + 4(n-1)]}{2} = 2n^2 + 7n\) (1 pt). \(S_{20} = 940 < 1000\) and \(S_{21} = 1{,}029 > 1000\): 21 terms (1 pt).

4 Finite geometric series / 4 pts

(a) \(a_1 = 7\), \(r = 2\) (1 pt): \(S_9 = 7(2^9 - 1) = 7\cdot 511 = 3{,}577\) (1 pt).

(b) \(a_1 = 4\), \(r = \tfrac12\), 6 terms (1 pt): \(S_6 = 4\cdot\dfrac{1 - \frac{1}{64}}{\frac12} = 8\cdot\dfrac{63}{64} = \dfrac{63}{8} = 7.875\) (1 pt).

5 Infinite series / 3 pts

(a) \(r = \dfrac13\) and \(|r| < 1\) (1 pt). \(S = \dfrac{18}{1 - \frac13} = 27\) (1 pt).

(b) \(r = 2\), and \(|r| \ge 1\), so the partial sums grow without bound and the series diverges (1 pt).

6 The pendulum / 2 pts

Geometric series with \(a_1 = 40\), \(r = 0.9\) (1 pt). \(S = \dfrac{40}{1 - 0.9} = 400\) cm, about 4 m (1 pt).