Question 1103117: Find the sum of the first 24 terms of the arithmetic series.
1+8+15+22+...
S(24)=
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
In ANY set of numbers, the sum of all the numbers is the average, multiplied by how many numbers there are.
In an arithmetic series, since the terms are "evenly spaced", the average of all the numbers is the average of the first and last terms.
So the sum of 24 terms of an arithmetic series is the average of the first and last terms, multiplied by 24.
The 24th term is the first term, plus the common difference added 23 times.
For this example, then...
The 24th term is 
The average of the first and last terms is 
And the sum of the first 24 terms is 
Note that many students prefer a different way of thinking of the sum of the terms of an arithmetic series. Instead of thinking
"(average of first and last) times (the number of terms)",
They group the numbers in pairs (1st term with 24th, 2nd with 23rd, and so on) so that each pair has the same sum; then they get the sum of all the terms in the series as
"(sum of first and last terms) times (the number of pairs)".
For your example, that method of thinking of the sum would give you...
There are 24 terms; so that is 12 pairs
The sum of the first and last terms is 1+162=163;
The sum of all the terms is 163*12 = 1956.
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