Question 1064992
Part (a):
————
a + (a+1) + (a+2) + … + (a+n-1)  
can be re-written:
 (a+a+ … + a) + (1+2+ … + n-1)

There are  n a's in the first parenthesis, and the 2nd parenthesis sums to  {{{ n*(n-1) / 2 }}} 

The sum S is:   {{{ highlight(S = an + n*(n-1)/2) }}}
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Part (b):
————
{{{ a*n + n*(n-1)/2 = 100 }}}
{{{ 2an + n(n-1) = 200 }}}
{{{ 2an + n^2 - n = 200 }}}
{{{  n^2 + 2an - n = 200 }}}
{{{  n(n + 2a - 1) = 200 }}}

a and n are integers, and both are > 0

We only need to try factors less than or equal to {{{ sqrt(200) }}}  which is approx 14.14,  an taking the floor of this value gives an max integer value of 14 to try.
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Factor 200 to see which factors we should try:
200 = 2*2*2*5*5
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We  want to try all combinations of the factors given above, such that their product is <= 14.
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Try n=2:    2(2+2a-1) = 200 —>   1+2a = 198   (won't work because 2a = 197  —> a is a non-integer)
Try n=4:    4(4+2a-1) = 200 —>  3+2a = 196 —>  2a = 193  (no good, a is non-integer)
Try n=8:    8(8+2a-1) = 200  —>  7+2a = 25 —> 2a = 18 —> a=9
                               {{{ highlight( a=9) }}}, {{{ highlight(n=8 )}}} satisfy the equality

n=5:   5(5+2a-1) = 200  —>  4+2a = 40 —> 2a = 36 —> a = 18
                  {{{ highlight(a=18) }}} ,  {{{ highlight(n=5) }}} satisfy the equality

n=10:   10(10+2a-1) = 200 —>  9+2a = 20  (no good, a is non-integer)

n=20:  n is too large  

I think that is it.