SOLUTION: the 5th term of an arithmetic sequence is 0 and the 13th term is equal to 12. Using suitable formulae determine the common difference,the first term,and the sum of the first 21 ter

Question 1069078: the 5th term of an arithmetic sequence is 0 and the 13th term is equal to 12. Using suitable formulae determine the common difference,the first term,and the sum of the first 21 terms
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The general form for the nth term of an arithmetic sequence is
:
a(n) = a(1) + d(n-1), where d is the common difference
:
we have two equations in two unknowns
:
1) a(5) = a(1) + d(5-1) = a(1) + 4d = 0
2) a(13) = a(1) + d(13-1) = a(1) + 12d = 12
:
solve equation 1) for a(1) and substitute for a(1) in equation 2
:
-4d + 12d = 12
:
*****************************************************************
d = 12/8 = 3/2
:
substitute for d in equation 1
:
a(1) + 4(3/2) = 0
a(1) = -6
:
check our answers, that is, d = 3/2 and a(1) = -6
:
equation 1
-6 + 4(3/2) = 0
0 = 0
:
equation 2
-6 +12(3/2) = 12
12 = 12
:
our answers for d and a(1) check
:
the sum of the first n terms of an arithmetic sequence is
:
S(n) = (1/2) * n * ( a(1) + a(n) )
:
we are asked for s(21)
:
note that a(21) = -6 + (3/2) * (21 - 1) = 24
:
S(21) = (1/2) * 21 * (-6 + 24) = 189
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