Practice: the arithmetic sequence

Sequences, grade 12 (4 units) Lesson 11 of 20Practice

Practice: the arithmetic sequence

1

In an arithmetic sequence $\displaystyle {a_1}={-7}$ and $\displaystyle {d}={4}$ .
Calculate $\displaystyle a_{31}$ .

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2

The arithmetic sequence: $\displaystyle 120, 113, 106, \ldots$

a

In which place is the first negative term?

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b

What is that term?

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3

In an arithmetic sequence $\displaystyle {a_6}={23}$ and $\displaystyle {a_{15}}={59}$ .
Find $\displaystyle a_1$ and $\displaystyle d$ .

4

Calculate the sum $\displaystyle 12+17+22+\ldots+252$ .

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5

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={2n^2+5n}$ .
Find $\displaystyle a_n$ , and prove that the sequence is arithmetic.

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6

The numbers $\displaystyle x+1$ , $\displaystyle 3x-2$ , $\displaystyle 4x+1$ (in this order) are three neighbouring terms of an arithmetic sequence.
Find $\displaystyle x$ and the three numbers.

7

A hall has two wings, and in both the seats are in rows.
In wing A the first row has one seat, and each further row has one seat more than the row before it.
In wing B the first row has 3 seats, and each further row has one seat more than the row before it.
Wing B has 32 rows more than wing A, and 25 times as many seats.
How many rows does wing A have?

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8

The price of a television is paid in 12 monthly payments, which form an arithmetic sequence.
The total is 1.52 times the sum of the first 6 payments.
The total is 1,900 ₪ more than the sum of the two middle payments.

a

Find the first payment and the difference of the sequence.

b

How much does the television cost?

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Formulas: sequences, grade 12 (4 units)

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
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$