Course 5.2Lesson 3 of 3

Arithmetic progressions in real life

Many everyday totals are sums of arithmetic progressions: money saved with a regular rise, seats in rows that grow, logs in a stack, prizes that fall by a fixed amount. The work is the same each time: find $a$ , $d$ and $n$

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Try the code and the simulations beside the lesson.

A method for word problems

Find the progression. What changes by the same amount each step? That gives $d$ .
Read $a$ , $d$ and $n$ , with their units.
Decide what is asked. A single term needs $a_n = a + (n - 1)d$ . A total needs $S_n$ .
Use the formula, then check that the answer fits the story.

Saving with a regular rise

Tanvi saves ₹ $40$ in the first week. Each week she saves ₹ $15$ more than the week before. How much has she saved after $10$ weeks?

The weekly amounts are $40, 55, 70, \ldots$ rupees, so $a = 40$ , $d = 15$ , $n = 10$ .
$S_{10} = 5[80 + 9 \times 15] = 5 \times 215 = 1075$ rupees.

Check. In the tenth week she saves $40 + 9 \times 15 = 175$ rupees. The average week is $\dfrac{40 + 175}{2} = 107.50$ rupees, and $10 \times 107.50 = 1075$ rupees.

Rows of seats

An open-air theatre has $18$ seats in the front row. Each row behind has $2$ more seats, and there are $25$ rows.

The last row has $18 + 24 \times 2 = 66$ seats.
The total is $\dfrac{25}{2}(18 + 66) = \dfrac{25}{2} \times 84 = 1050$ seats.

A stack of logs

Logs are stacked so that the bottom row has $20$ logs, the next $19$ , then $18$ , and so on. The top row has $6$ logs.

Here $a = 20$ , $d = -1$ and $l = 6$ . The number of rows is $n = 20 - 6 + 1 = 15$ .
The total is $\dfrac{15}{2}(20 + 6) = 15 \times 13 = 195$ logs.

A negative $d$ is fine. The progression goes down instead of up.

Prizes that fall

A school gives $6$ prizes worth ₹ $1950$ in all. Each prize is ₹ $40$ less than the one before it. What is the first prize?

\displaystyle \frac{6}{2}[2a + 5 \times (-40)] = 1950 \quad \Rightarrow \quad 2a + 5 \times (-40) = 650 \quad \Rightarrow \quad 2a = 850.

So $a = 425$ , and the first prize is ₹ $425$ .

The prizes are ₹ $425$ , ₹ $385$ , ₹ $345$ , ₹ $305$ , ₹ $265$ and ₹ $225$ . They add up to ₹ $1950$ .

Practise with new numbers

Try it in Python

The program prints Tanvi's savings week by week, and compares the last total with the formula.

Common mistakes

Answering with a term when the total is asked. "How much in the tenth week" needs $a_{10}$ . "How much in ten weeks" needs $S_{10}$ .
Counting rows wrongly. From $20$ down to $6$ there are $15$ rows, not $14$ .
Dropping the units. A sum of rupees is in rupees. A sum of seats is a number of seats.
Forgetting that $d$ can be negative. Prizes or rows that get smaller have $d < 0$ .

Remember these

Find what grows by a fixed amount: that is $d$ .
A single amount is a term, $a_n$ . A total is a sum, $S_n$ .
When the first and last amounts are known, $S_n = \dfrac{n}{2}(a + l)$ is quickest.
Check the answer against the story.