Question 1110298
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Since the sum of the 3rd and 5th terms of the AP is 20, the 4th term is 10.<br>
If the common difference in the AP is d, then the 2nd term is 10-2d and the 8th term is 10+4d.  Since the 2nd, 4th, and 8th terms of the AP form a GP,
{{{10/(10-2d) = (10+4d)/10}}}
{{{100+20d-8d^2 = 100}}}
{{{20d-8d^2 = 0}}}
{{{4d(5-2d) = 0}}}
{{{d = 0}}}  or  {{{d = 2.5}}}<br>
Both values of d satisfy the conditions of the problem; but the AP with d=0 is not very interesting.<br>
With d=2.5 and the 4th term=10, the AP is
2.5, 5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, 25, ...<br>
Answers:
a) 2.5, 5, 7.5, 10
b) (number of terms) times (average of first and last terms) = {{{10((2.5+25)/2) = 5*27.5 = 137.5}}}