What is the sum when the ratio is 1?

Every term equals a_1, so S_n = na_1. The formula \frac{a_1(r^n - 1)}{r - 1} only works for r \neq 1.

Math B

Arithmetic and Geometric Sequences: Practice Problems

Arithmetic and geometric sequences are the foundation of the topic. Practise the formulas for $a_n$ and $S_n$ and how to find the first term, difference or ratio from given conditions.

Basic, Standard, Advanced: Grade 11 · Term 2

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Arithmetic sequences

A sequence with a constant difference $d$ between consecutive terms is arithmetic.

Formulas

a_n = a_1 + (n-1)d

S_n = \frac{1}{2}n(a_1 + a_n) = \frac{1}{2}n\{2a_1 + (n-1)d\}

Example: with first term $3$ and difference $4$ , $a_n = 4n - 1$ and $S_{10} = \frac{1}{2} \cdot 10 \cdot (3 + 39) = 210$ .

Arithmetic mean

$a, b, c$ form an arithmetic sequence in this order $\iff 2b = a + c$ .

Geometric sequences

A sequence with a constant ratio $r$ between consecutive terms is geometric.

Formulas

a_n = a_1r^{n-1}

S_n = \frac{a_1(r^n - 1)}{r - 1} = \frac{a_1(1 - r^n)}{1 - r}\quad (r \neq 1),\qquad S_n = na_1\quad (r = 1)

Geometric mean

$a, b, c$ form a geometric sequence in this order $\iff b^2 = ac$ (so $b$ may have two values, $\pm$ ).

Finding a sequence from conditions

Two given terms: let the first term be $a$ and the difference $d$ (or ratio $r$ ), and solve the simultaneous equations. For geometric sequences, dividing one equation by the other eliminates $a$ .
Maximum sum: if the first term is positive and the difference negative, the sum is largest once all positive terms are added. Find the $n$ with $a_n > 0$ .
Common terms: the numbers shared by two arithmetic sequences recur every lcm of the two differences.
Three numbers: write them as $a - d,\ a,\ a + d$ (arithmetic) or $\frac{a}{r},\ a,\ ar$ (geometric).

Savings plans

Depositing $a$ at the start of every year with annual interest rate $r$ (compounded yearly), the total at the end of year $n$ is

a(1 + r) + a(1 + r)^2 + \cdots + a(1 + r)^n = \frac{a(1 + r)\{(1 + r)^n - 1\}}{r},

a geometric sum: earlier deposits earn interest for more years.

Q1BasicGr 11 · Term 2Arithmetic and geometric sequences

Find the general term $a_n$ of the arithmetic sequence $9,\ 10,\ 11,\ 12,\ \ldots$ .

Answer

a_n = n + 8

Solution

Read off the first term and the common difference.
$a_1 = 9,\quad d = 10 - 9 = 1$
Substitute into $a_n = a_1 + (n-1)d$ .
$\begin{aligned}a_n &= 9 + (n - 1) \cdot 1 \\ &= n + 8\end{aligned}$

Q2StandardGr 11 · Term 2Arithmetic and geometric sequences

You save $200$ yen in the first month and, from the second month on, $100$ yen more than in the previous month. Find the total amount saved in $12$ months.

Answer

9000\text{ yen}

Solution

Sum the first $12$ terms of the arithmetic sequence with first term $200$ and difference $100$ .
$\frac{1}{2} \cdot 12 \cdot \{2 \cdot 200 + 11 \cdot 100\} = 9000$

Q3AdvancedGr 11 · Term 2Arithmetic and geometric sequences

Three numbers form an arithmetic sequence. Their sum is $-9$ and their product is $21$ . Find the three numbers.

Answer

-7,\ -3,\ 1

Solution

Let the numbers be $a - d,\ a,\ a + d$ . From the sum,
$3a = -9 \quad\Longrightarrow\quad a = -3$
From the product,
$-3(9 - d^2) = 21 \quad\Longrightarrow\quad d^2 = 16,\ d = \pm 4$
Either sign gives the same three numbers.
$-7,\ -3,\ 1$

Q1BasicGr 11 · Term 2Arithmetic and geometric sequences

For the arithmetic sequence $-9,\ -14,\ -19,\ \ldots$ , answer the following.

(1)Find the 14th term.
(2)Which term is $-109$ ?

Q2StandardGr 11 · Term 2Arithmetic and geometric sequences

You save $100$ yen in the first month and, from the second month on, $50$ yen more than in the previous month. Find the total amount saved in $10$ months.

Q3AdvancedGr 11 · Term 2Arithmetic and geometric sequences

Three numbers form a geometric sequence. Their sum is $13$ and their product is $27$ . Find the three numbers.

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Sigma notation and sums of sequences

Recurrence relations and induction