Practice: the geometric sequence

Sequences, grade 12 (4 units) Lesson 18 of 20Practice

Practice: the geometric sequence

1

In a geometric sequence $\displaystyle {a_1}={4}$ and $\displaystyle {q}={3}$ .
Calculate $\displaystyle a_6$ .

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2

In a geometric sequence $\displaystyle {a_2}={-6}$ and $\displaystyle {a_5}={48}$ .
Find $\displaystyle q$ and $\displaystyle a_1$ .

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3

The sequence: $\displaystyle {a_n}={7\cdot5^{n}}$ .
Prove that the sequence is geometric, and find its ratio.

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4

The geometric sequence: $\displaystyle 64, 32, 16, \ldots$
Calculate the sum of the first 6 terms.

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5

The numbers $\displaystyle x+1$ , $\displaystyle x+5$ , $\displaystyle x+17$ (in this order) are three neighbouring terms of a geometric sequence.
Find $\displaystyle x$ and the ratio.

\displaystyle q

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6

The sum of an infinite geometric sequence is 200, and its first term is 50.
Find the ratio.

7

The infinite geometric sequence: $\displaystyle 48, 24, 12, \ldots$
Calculate the sum of the terms in the odd places.

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8

An app was downloaded 200 times in its first week.
Each week it was downloaded 3 times as often as the week before.
How many downloads were there in all in the first 6 weeks?

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9

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={5(2^n-1)}$ .
Find $\displaystyle a_n$ , and explain why the sequence is geometric.

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