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Core 2: Sequences and Series

This particular section has examples which cover the main types of questions you will meet in the examination. The focus will be on arithmetic and geometric sequences.

 

Arithmetic sequence

 

 

The terms progress by a common difference.

 

 

1, 5, 9, 13, 17

 

This is arithmetic:

a=1 (first term)

d=+4 (the difference)

 

 

7, 2, -3, -8

 

Also arithmetic

a=7

d=-5 (be careful to get the minus sign correct)

 

 

Given an arithmetic sequence, it is possible to find a general expression for the kth term

 

 

 

Learn this:

 

 

 

Given the sequence:

7, 13, 19, 25, ...

Find an expression for the kth term and hence the 101st term.

 

 

Is 1034 in the sequence:

6, 19, 32, 45,...

 

Need to find k such that

 

This is not a whole number, so 1034 is not in the sequence

 

Finding the sum of an arithmetic sequence

 

 

 

 

If you know that first and last term of a sequence and the number of terms there is a simple formula:

 

 

If you know the first term, the number of terms and the common difference then there is a slightly more complicated formula:

 

 

 

Example: Find the sum of first 50 even numbers.

2+4+6+...+100

 

 

Use the first formula

 

 

Find the sum of the first 100 terms of the sequence:

 

3, 7, 11, 15,...

 

 

 

 

 

 

Finding the number of terms required to get to a particular sum.

 

 

 

How many terms are required for the sum of the following sequence to be over 1000

 

3, 5, 7, 9,...

 

 

 

 

 

 

 

 

 

Need positive answer

 

Round up:

Need at least 31 terms.

 

 

Geometric sequence

 

 

The terms progress by a constant ratio (multiplier)

 

5,10,20,40,...

 

 

18,6,2,...

 

 

 

Formula for the kth term

 

 

 

Find a general expression for the kth term in the series:

6,12,24,...

 

 

When is the sequence greater than 300,000?

 

 

 

 

 

 

Need:

 

Divide by 6

 

 

Take logarithms

 

Need at least 17 terms

 

 

There is a formula for finding the sum of the first n terms of a geometric sequence.

 

 

 

Find the sum of the first 20 terms of the sequence

6, 12, 24, ...

 

 

 

Find when the sum is greater than 300,000,000

 

 

 

 

 

 

Need:

Round up:

Need at least 26 terms for the sum to be greater than 300,000,000.

 

If the ratio is between -1 and 1 you can find the sum to infinity.

This is the hypothetical sum if you were able to add together all the terms.

Because the terms get smaller and smaller when r is between -1 and 1, the sum approaches a limit.

 

 

 

Find the sum to infinity of the sequence:

 

10,5,2.5,...

 

 

 

 

 

The sum to infinity of a sequence is 40 and the second term is 8. Find a and r.

 

 

 

Re-arrange (1)

 

 

Substitute in (3)

 

 

Divide by 8

 

or

 

Both options are a valid solution.