Question 1028709
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find the twentieth term of the arithmetic sequence in which t4 = a and t12=b?
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In other words, find the term  {{{t[20]}}}  of an AP,  if you are given that
{{{t[4]}}} = a   and  {{{t[12]}}} = b.


You have 

{{{t[4]}}} = {{{t[1] + 3d}}} = a      and
{{{t[12]}}} = {{{t[1] + 11d}}} = b.

Consider  {{{2*t[12]}}} - {{{t[4]}}} = {{{2*(t[1] + 11d) - (t[1]+3d)}}} =  {{{2t[1] + 22d - t[1] - 3d}}} = {{{t[1] + 19d}}}.

The right side is exactly  {{{t[20]}}}.

Therefore,  {{{t[20]}}} = 2b - a.

<U>Answer</U>.  {{{t[20]}}} = 2b - a.
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