📝 Chapter 5: Sequences and Series

An arithmetic sequence has \( a = 8, d = -3 \). Find \( t_{10} \).

Find \( t_5 \) for the geometric sequence with \( a = 4, r = -2 \).

Find the sum of the first 30 terms of \( 4 + 9 + 14 + 19 + \ldots \).

Find \( S_8 \) for the geometric series \( 6 + 12 + 24 + \ldots \).

Identify whether each sequence is arithmetic, geometric, or neither: (a) 1, 4, 9, 16; (b) 5, 10, 20, 40; (c) 12, 9, 6, 3. Justify each.

Evaluate \( \displaystyle\sum_{k=1}^{5}(3k - 1) \).

An arithmetic sequence has \( t_4 = 14 \) and \( t_9 = 39 \). Find \( a \), \( d \), and the general term \( t_n \).

Find \( S_n \) for the geometric series \( 5 - 10 + 20 - 40 + \ldots \) with \( n = 8 \). Show all work.

Find the number of terms \( n \) so that \( 2 + 5 + 8 + \ldots + t_n = 175 \). Show all reasoning.

Compare and contrast arithmetic and geometric sequences. State the test you'd use to identify each, and give one example of each.

Derive the formula \( S_n = \tfrac{n}{2}(2a + (n-1)d) \) using the Gauss "pair-up" trick. Use \( 1 + 2 + \ldots + 100 \) as your worked example.

Express \( 3 + 6 + 12 + 24 + 48 \) in sigma notation. Then explain why this matches the geometric series sum formula.

Translate the recursive definition \( t_1 = 5, t_n = 2 t_{n-1} + 1 \) into the first 4 terms. Then explain why this is neither arithmetic nor geometric.

A theatre has 25 rows. Row 1 has 18 seats; each subsequent row has 2 more seats than the previous. (a) How many seats in row 25? (b) How many seats total?

A pendulum swings 80 cm on its first arc. Each subsequent arc is 90% of the previous. (a) Find the length of the 6th arc. (b) Find the total distance travelled in the first 10 arcs.

Saved \$50 in week 1, \$60 in week 2, \$70 in week 3, etc. (a) How much in week 20? (b) Total saved over 20 weeks? (c) After how many weeks does the cumulative total first exceed \$2000?