In an arithmetic sequence, $t_5 = 23$ and $t_{12} = 58$. Find $a$, $d$ and $t_{40}$.

$a = 3$, $d = 5$, $t_{40} = 198$

A theatre has $24$ rows. The first row has $18$ seats and each row has $2$ more seats than the row in front of it. How many seats are there, and how many rows would it need to seat at least $1000$ people?

$984$ seats; $25$ rows to seat at least $1000$.

Grade 11 · MCR3U · Functions

MCR3U: Arithmetic Sequences and Series

An arithmetic sequence goes up or down by the same amount at every step. Two formulas cover the topic: one gives any term without listing the ones before it, and one adds up as many terms as you like.

How It Works

Arithmetic sequences. Each term is the one before plus a common difference $d$ : $5, 8, 11, 14, \dots$ has first term $a = 5$ and $d = 3$ . To check a sequence, subtract each term from the next. If every difference is the same, it is arithmetic.

The general term. $t_n = a + (n - 1)d$ . To get to term $n$ you add $d$ only $n - 1$ times, because the first term is already there. The same sequence as a recursive formula is $t_1 = a$ and $t_n = t_{n - 1} + d$ .

How many terms. Set the general term equal to the last term and solve for $n$ . For $5, 8, \dots, 152$ : $5 + 3(n - 1) = 152$ , so $n - 1 = 49$ and $n = 50$ .

Arithmetic series. A series is the sum of the terms of a sequence. Write the sum forward and backward and add the two lines: each pair of terms adds to $a + t_n$ , and there are $n$ pairs, so $2S_n = n(a + t_n)$ . That gives $S_n = \frac{n}{2}(a + t_n)$ , or with $t_n$ written out, $S_n = \frac{n}{2}\left(2a + (n - 1)d\right)$ . Use the first form when you know the last term and the second when you know $d$ .

Two terms given. If you know two terms but not $a$ or $d$ , write each one with the general term and solve the pair of equations. Subtracting them removes $a$ and leaves $d$ .

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
For the sequence $7, 11, 15, 19, \dots$ find $t_{20}$ .
Show solution
1. $a = 7$ and $d = 11 - 7 = 4$ .
2. $t_{20} = 7 + (20 - 1)(4) = 7 + 76$ .
Answer: $t_{20} = 83$
Easy
How many terms are in the sequence $-4, 2, 8, \dots, 152$ ?
Show solution
1. $a = -4$ and $d = 6$ .
2. $-4 + 6(n - 1) = 152$ , so $6(n - 1) = 156$ and $n - 1 = 26$ .
Answer: $27$ terms
Medium
Find the sum of the first $30$ terms of $3 + 10 + 17 + \dots$
Show solution
1. $a = 3$ , $d = 7$ and $n = 30$ .
2. $S_{30} = \frac{30}{2}\left(2(3) + 29(7)\right) = 15(6 + 203) = 15(209)$ .
Answer: $S_{30} = 3135$
Medium
In an arithmetic sequence, $t_5 = 23$ and $t_{12} = 58$ . Find $a$ , $d$ and $t_{40}$ .
Show solution
1. $a + 4d = 23$ and $a + 11d = 58$ .
2. Subtract: $7d = 35$ , so $d = 5$ and $a = 23 - 4(5) = 3$ .
3. $t_{40} = 3 + 39(5) = 198$ .
Answer: $a = 3$ , $d = 5$ , $t_{40} = 198$
Test level
A theatre has $24$ rows. The first row has $18$ seats and each row has $2$ more seats than the row in front of it. How many seats are there, and how many rows would it need to seat at least $1000$ people?
Show solution
1. The rows form an arithmetic sequence with $a = 18$ and $d = 2$ .
2. $S_{24} = \frac{24}{2}\left(2(18) + 23(2)\right) = 12(82) = 984$ .
3. For $n$ rows, $S_n = \frac{n}{2}\left(36 + 2(n - 1)\right) = n(n + 17)$ . Solve $n^2 + 17n = 1000$ : $n = \frac{-17 + \sqrt{4289}}{2} \approx 24.25$ .
4. $n$ is a whole number, so round up and check: $S_{25} = 25(42) = 1050 \ge 1000$ , while $S_{24} = 984$ falls short.
Answer: $984$ seats; $25$ rows to seat at least $1000$ .

Learn It on Martian Lab

Martian Lab teaches this in the Arithmetic Sequences lesson of its MCR3U Discrete Functions unit. Watch it, then take the Discrete Functions practice tests to see if it stuck.

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Free Discrete Functions practice test, with solutions

More Discrete Functions topics: Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

More MCR3U topics: Inverse Functions · Transformations of Functions · Sine Law and the Ambiguous Case · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Special Angles and the CAST Rule · Cosine Law and 3D Problems · Transformations of Sine and Cosine · Sinusoidal Word Problems

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