SOLUTION: The ninth term of an arithmeticsequence is 42 and the sum of the fist term is 414. Find the first term of the sequence.

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Question 845973: The ninth term of an arithmeticsequence is 42 and the sum of the fist term is 414. Find the first term of the sequence.
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
Let the first term of the AP be 'a' and the common difference be 'd'.
Therefore the n-th term is t%5Bn%5D=a%2B%28n-1%29d
The sum up to first n-terms is S%5Bn%5D=n%282a%2B%28n-1%29d%29%2F2

Here, n=9,

9th term is given by
t%5B9%5D=42
a%2B%289-1%29d+=+42
a%2B8d=42 _______ (1)

Sum of first 9 terms is given by
S%5B9%5D=414
9%282a%2B%289-1%29d%29%2F2=414
9%282a%2B8d%29%2F2=414
9%28a%2B4d%29=414
a%2B4d=414%2F9
a%2B4d=46 _______ (2)

Subtracting both sides of equation (2) from (1)
4d=-4
d=-1

Substituting d=-1 in (2) we have
a-4=46
a=50

First term of the sequence is 50.
Cheers!!!