SOLUTION: given a1=28 an=-77 and sn =-196 in an A.P find the 10th term and the sum of the first 10 terms

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Question 1072952: given a1=28 an=-77 and sn =-196 in an A.P find the
10th term and the sum of the first 10 terms
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
a%5B1%5D=28                <--first term
a%5Bn%5D=a%5B1%5D%2B%28n-1%29d=-77   <--formula for nth term
a%5Bn%5D=28%2B%28n-1%29d=-77
28%2B%28n-1%29d=-77
%28n-1%29d+=+-105

S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29=+-196  <-- formula for nth term

 Substitute for a1 and (n-1)d=-105

expr%28n%2F2%29%282%2828%29-105%29=+-196

expr%28n%2F2%29%2856-105%29=+-196

expr%28n%2F2%29%28-49%29=+-196

Multiply through by 2

n%28-49%29=+-392

n=%28-392%29%2F%28-49%29

n=8          <--number of terms whose sum is -77

We'll need d, so

%28n-1%29d=-105
%288-1%29d=-105
7d=-105
d=-15        <-- common difference, d

To find 10th term:

a%5Bn%5D=a%5B1%5D%2B%28n-1%29d

Substitute n=10, a1 = 28, n=10, d=-15

a%5B10%5D=28%2B%2810-1%29%28-15%29

a%5B10%5D=28%2B9%28-15%29

a%5B10%5D=28-135

a%5B10%5D=-107     <-- 10th term

S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29 <-- formula for sum of first n terms

S%5B10%5D=expr%2810%2F2%29%282%2828%29%2B%2810-1%29%28-15%29%29

S%5B10%5D=%285%29%2856%2B%289%29%28-15%29%29

S%5B10%5D=%285%29%2856-135%29

S%5B10%5D=%285%29%28-79%29

S%5B10%5D=-395

Checking, the sequence is:

28,13,-2,-17,-32,-47,-62,-77,-92,-107

Add them

28+13-2-17-32-47-62-77-92-107 = -395

Add the first 8 terms

28+13-2-17-32-47-62-77 = -196

Everything checks.

Edwin