Question 1094750
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A key concept for working with arithmetic progressions is that we can group the terms in pairs, with each pair having the same sum.<br>
In your example, where the sum of the first 20 terms is 840, we know those 20 terms are 10 pairs, so the sum of each pair is 840/10 = 84.<br>
One of the pairs is the first and last (20th) terms.  The 20th term is the first term, plus the common difference (which is 4) 19 times; and the sum of the first and 20th terms is 84.  So
{{{a + (a+19(4)) = 2a+76 = 84}}}  -->  a = 4.<br>
That's the answer to part (a).<br>
Part (b): In finding the sum of the first 10 terms, we will have 5 pairs with the same sum; one of those pairs is the 1st and 10th terms.  The 10th term is the first term 4, plus the common difference (4) 9 times; 4+9(4) = 40.  So the sum of each of the 5 pairs is 4+40 = 44; since there are 5 pairs, the sum of the first 10 terms is 44*5 = 220.