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In an arithmetic sequence...

the third term is 10 and the fifth term is 16.
 
a) find the common difference
 
b) find the first term
 
c) find the sum the first 20th term of the sequence
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1 Answer

in an arithmetic series, an = a0+ (n-1) *d
 
that means the nth term is equal to the first term plus n-1 times the common difference.
 
In this case we are told a3=10 and a5=16
 
so:  a3= a+ 2*d =10    => a0= 10-2d     
and      
      a5= a+4*d = 16     => a0= 16-4d
 
so 10-2d = 16-4d => 2d=6 => d=3
 
now you can plug this value of d into any of those equations we got for a0 and find a0.
for example a0= 10-2(3) = 4 so the first term is 4
 
the formula for the sum of the first n terms is:
 
Sn = (n/2)* [2a0+ (n-1)*d]
 
so S20 = (20/2) * [2*4 + (19)*(3)] = 10* [65] = 650