Word problems with arithmetic sequences

Sequences, grade 12 (4 units) Lesson 10 of 20Learning

Word problems with arithmetic sequences

Example: saving

Noa saves: 50 ₪ in the first month, and each month 10 ₪ more than in the month before.
How much will she save in a year?

The monthly amounts are an arithmetic sequence: $\displaystyle {a_1}={50}$ , $\displaystyle {d}={10}$ , $\displaystyle {n}={12}$ .
$\displaystyle {S_{12}}={\frac{12(2\cdot50+11\cdot10)}{2}}={\frac{12\cdot210}{2}}={1260}$ ₪.

A sequence in a word problem

In every such problem three questions decide what to work out:

what the terms are: seats in a row, a payment a month, a distance a day;
what is given: $\displaystyle a_1$ , $\displaystyle d$ , the number of terms, a certain term or a sum;
what is asked: a term ( $\displaystyle a_n$ ), a place ( $\displaystyle n$ ) or a sum ( $\displaystyle S_n$ ).

Example: a training plan

Dana runs 2 km on the first day, and each day 0.5 km more than the day before.
On which day will she run 10 km?

The distances: $\displaystyle {a_1}={2}$ , $\displaystyle {d}={0.5}$ ; the question is the place of the term 10.
$\displaystyle {2+(n-1)\cdot0.5}={10}$ , so $\displaystyle {(n-1)\cdot0.5}={8}$ , and $\displaystyle {n-1}={16}$ .
On day 17.

Exercise 1

A pile of pipes has 20 pipes in the bottom row.
Each row above it has one pipe fewer, and the top row has one pipe.
How many pipes are in the pile?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A loan is paid back in 12 monthly payments, which form an arithmetic sequence.
The first payment is 500 ₪, and the last is 830 ₪.

How much was paid back in all?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

By how many shekels does the payment grow each month?

Exercise 3

A basketball player practises shooting.
On the first day he shoots 40 times, and each day 5 times more than the day before.

On which day will he shoot 135 times?

After how many days will the total number of shots reach 1,125?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: sequences, grade 12 (4 units)

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A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$