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Question 257916: In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sum of the terms of an arithmetic series is given by the formula:
s[n] = n * (a[1] + a[n])/2
s[n] = sum of the n terms of the sequence.
a[1] = first term in the sequence.
a[n] = nth term in the sequence.
n = number of terms in the sequence.
the formuula for the nth term in the sequence is:
a[n] = a[1] + (n-1) * d
d is the common difference.
you will use both these formulas to find the answer.
you are given:
sum of the first 9 terms in a sequence is 162
sum of the first 12 terms in the same sequence is 288.
the formnula for the sum of the first n terms in a sequence is:
s[n] = n * (a[1] + a[n])/2
substitute for the first 9 terms to get:
162 = 9 * (a[1] + a[9])/2 (first equation)
substitute for the first 12 terms to get:
288 = 12 * (a[1] + a[12])/2 (second equation)
multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get:
a[1] + a[9] = 36 (third equation)
multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get:
a[1] + a[12] = 48 (fourth equation)
subtract third equation from fourth equation to get:
a[12] - a[9] = 12
since 12 - 9 = 3, there are 3 terms between a[9] and a[12].
12/3 = 4 which means the common difference between each term is 4.
the nth term in each sequence is given by the formula:
a[n] = a[1] + (n-1) * d
we now know that d = 4
we solve for a[9] to get:
a[9] = a[1] + 8*4 = a[1] + 32
we take our first equation and plug in our new found values.
first equation is:
162 = 9 * (a[1] + a[9])/2 (first equation)
substitute a[1] + 32 for a[9] to get:
162 = 9 * (a[1] + a[1] + 32)/2
combine like terms to get:
162 = 9 * (2* a[1] + 32)/2
multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get:
36 = 2 * a[1] + 32
subtract 32 from both sides of this equation to get:
36-32 = 2 * a[1] which becomes:
4 = 2 * a[1].
divide both sides of this equation by 2 to get:
a[1] = 2.
we now know a[1] and we know d so we can construct the sequence.
formula for nth term in the sequence is:
a[n] = a[1] + (n-1) * d
since d = 4,...
the 9th term in the sequence is:
a[9] = 2 + 8*4 = 2 + 32 = 34
the 12th term in the sequence is:
a[12] = 2 + 11*4 = 2 + 44 = 46
the sequence is:
term: 1 2 3 4 5 6 7 8 9 10 11 12
sequence: 2 6 10 14 18 22 26 30 34 38 42 46
2 is the 1st term.
34 is the 9th term.
46 is the 12th term.
common difference is 4.
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