1.2 - Arithmetic and Geometric Sequences and Series - Completed Notes

non zero

A sequence is said to be geometric if each term can be found by multiplying the previous term by a(n) ________ constant.

arithmetic sequence

Ex: Three numbers are consecutive terms of a(n) ________.

Ex

Consider the sequence 2, 9, 16, 23, 30, … (a) Show that the sequence is arithmetic (b) Find a formula for the general term (c) Find the 100th term of the sequence (d) Is (i) 828 and (ii) 2341 a term of the sequence

Ex

Find k given that and are consecutive terms of an arithmetic sequence

Ex

Find the general term for an arithmetic sequence with and

Ex

Insert four numbers between 2 and 17 so that all six numbers are in an arithmetic sequence

Ex

Ryan is a cartoonist

Ex

Three numbers are consecutive terms of an arithmetic sequence

We know that the terms of the series are

and these can be written in terms of the first term and the common difference

We can then reverse the terms and write them in terms of the last term, which gives us

If we add the two equations on the previous page this gives us

Ex

Find the sum of 2 + 9 + 16 + 23 + … to 30 terms

Ex

The first term of an arithmetic series is -7 and the fourth term is 23

Ex

The sum of an arithmetic series is given by Find the common difference and the first three terms of the series

Examples

2, 4, 8, 16, …

Ex

Consider the sequence , , (a) Show that the sequence is geometric (b) Find the general term, (c) Hence, find the 12th term

Ex

are consecutive terms of a geometric sequence

Ex

Find the number of terms in the geometric sequence

Ex

A geometric sequence has and

Ex

Find the first term of the sequence which exceeds 1400

Ex

The initial population of rabbits was 50

(a) How many rabbits were present after

(i) 15 weeks (ii) 30 weeks (b) How long would it take for the population to reach 500

Ex

$5000 is invested at 7% for 4 years compounded annually

Ex

The fourth term of a geometric sequence is 54 and the sixth term is 486

If we have a special case where all terms are the same and thus, Ex

Find the sum of the series

Ex

Find the sum of the first 5 terms of a geometric series with a first term of 3 and a common ratio of 2

Ex

Find two possible geometric sequences where the sum of the first two terms is 20 and the sum of the first four terms is 1640

Ex

The sum of the first n terms of a geometric sequence is given by

Ex

Determine how many terms are required for the sum of the geometric series given by to exceed 1000

Ex

Find the sum of

Ex

For what values of x does the series converge

Ex

A geometric series converges to 8