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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 k^{th}
term 
Learn this: 
Given the sequence: 7, 13, 19, 25, ... Find an expression for the k^{th}
term and hence the 101^{st} 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 k^{th}^{
}term 

Find a general expression for the k^{th}
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. 
Rearrange (1) Substitute in (3) Divide by 8 or Both options are a valid solution. 