Sequences, Series, and Induction: math practice, Grade 12 – download the PDF

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7 October 20267 October 2026

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Math practice Grade 12 : Sequences, Series, and Induction — Zyro the alien explorer of Planète Maths

21 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Terms of an arithmetic sequence ★★★

An arithmetic sequence has \(a_1 = 4\) and common difference \(d = 6\). List the first five terms and find \(a_{12}\).

2 Arithmetic, geometric, or neither? ★★★

Classify each sequence and justify: (a) 3, 9, 27, 81, ... (b) 50, 43, 36, 29, ... (c) 2, 5, 10, 17, ...

3 Expand a sum ★★★

Write out and evaluate \(\displaystyle\sum_{k=1}^{5} (3k - 2)\).

4 Fifth term of a geometric sequence ★★★

A geometric sequence has \(a_1 = 2\) and \(r = 5\). Find \(a_5\).

5 Does the series have a sum? ★★★

True or false, with justification: (a) \(3 + 6 + 12 + 24 + \cdots\) has a finite sum. (b) \(5 + 1 + 0.2 + 0.04 + \cdots\) has a finite sum. If it does, find it.

6 Recursive to explicit ★★★

A sequence is defined by \(a_1 = 5\) and \(a_n = a_{n-1} + 8\) for \(n \ge 2\). Write the first four terms and give an explicit formula for \(a_n\).

7 Expand with Pascal’s triangle ★★★

Use row 5 of Pascal’s triangle to expand \((x+1)^5\).

8 Stadium seating ★★★

The first row of a small stadium section has 22 seats, and each row behind it has 3 more seats than the previous one. There are 18 rows. How many seats are in the last row, and how many seats are in the section?

9 Weekly savings ★★★

You save 50 dollars in week 1 and increase your deposit by 5 dollars every week. What do you deposit in week 26, and how much have you saved in total after 26 weeks?

10 A spreading rumor ★★★

On day 1, 6 students hear a rumor. Each following day, twice as many students newly hear it as on the previous day. How many students have heard it in total after 10 days?

11 How many terms? ★★★

The geometric sequence 3, 12, 48, ... ends with the term 3072. (a) Find the number of terms. (b) Find the sum of all the terms.

12 Repeating decimal as a fraction ★★★

Write \(0.454545\ldots\) (repeating 45) as an infinite geometric series and find the fraction it equals.

13 Using the sum rules ★★★

Use the formulas for \(\sum k\) and \(\sum k^2\) to evaluate: (a) \(\displaystyle\sum_{k=1}^{40} (3k + 2)\); (b) \(\displaystyle\sum_{k=1}^{12} k^2\).

14 A Fibonacci-style sequence ★★★

Let \(f_1 = 1\), \(f_2 = 3\), and \(f_n = f_{n-1} + f_{n-2}\) for \(n \ge 3\). Find \(f_9\) and the first index \(n\) for which \(f_n \gt 50\).

15 Two coefficients in a binomial expansion ★★★

(a) Find the coefficient of \(x^3\) in the expansion of \((2x - 1)^6\). (b) Find the constant term in the expansion of \(\left(x^2 + \dfrac{1}{x}\right)^6\).

16 Induction: a sum of products ★★★

Prove by induction that \(1\cdot 2 + 2\cdot 3 + 3\cdot 4 + \cdots + n(n+1) = \dfrac{n(n+1)(n+2)}{3}\) for all \(n \ge 1\).

17 Induction: divisibility ★★★

Prove by induction that \(4^n - 1\) is divisible by 3 for every integer \(n \ge 1\).

18 Find the formula, then prove it ★★★

A sequence satisfies \(a_1 = 2\) and \(a_n = 2a_{n-1} + 3\) for \(n \ge 2\). (a) Compute \(a_2, a_3, a_4\). (b) Show that \(a_n = 5\cdot 2^{n-1} - 3\) by induction.

19 The bouncing ball ★★★

A ball is dropped from a height of 10 feet. After each impact it rebounds to 60% of the height it fell from. Find the total distance the ball travels before coming to rest. Give the answer in feet, then convert to meters (1 ft = 0.3048 m).