SOLUTION: The sum of an AP to four terms is 38. The sum of the same AP to eight terms is 124. How do I work out the first term and the common difference?

Question 1004534: The sum of an AP to four terms is 38.
The sum of the same AP to eight terms is 124.
How do I work out the first term and the common difference?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!

you are given that sum of 4 terms is 38 and sum of 8 terms is 124.

the formula for sum of an arithmetic sequence is Sn = n/2 * (A1 + An)

the formula for the nth term of an arithmetic sequence is An = A1 + (n-1)d

A1 is the first term.
An is the nth term.
n is the number of terms.
d is the common difference.

since An is equal to A1 + (n-1)d, you can replace An in the formula of Sn = n/2 * (A1 + An with A1 + (n-1)d to get:

Sn = n/2 * (A1 + An) becomes:

Sn = n/2 * (A1 + A1 + (n-1)d) which then becomes:

Sn = n/2 * (2A1 + (n-1)d)

it is this last formula that will help you find what you are looking for.

you are given that the sum of the first 4 terms in the sequence is equal to 34.

the summation formula for that becomes:

38 = 4/2 * (2A1 + 3d)

you can simplify this formula to get:

38 = 4A1 + 6d

you are also given tha the sum of the first 8 terms in the sequence is equal to 124.

the summation formula for that becomes:

124 = 8/2 * (2A1 + 7d)

You can simplify this to get:

124 = 8A1 + 28d

you now have 2 formulas that you need to solve simultaneously.

they are:

38 = 4A1 + 6d
124 = 8A1 + 28d

multiply both sides of the first equation by 2 and bring down the second equation as is to get:

76 = 8A1 + 12d
124 = 8A1 + 28d

subtract the first equation from the second equation to get:

48 = 16d

solve for d to get:

d = 3

now that you know d = 3, you can use either original equation to solve for A1.

your original equations to use are:

38 = 4A1 + 6d
124 = 8A1 + 28d

you will get A1 = 5.

your solution is that A1 = 5 and d = 3

A1 is the first term.
d is the common difference.

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