Progressions!

The sum of the terms of the arithmetic progression is 93, then if we call these terms a-d,a and a+d, where d is the difference of the arithmetic progression we have a-d + a + a+d = 3a= 93 $\Rightarrow$ a= 31(the 2nd term in the arithmetic progression is equal to 31) $\Rightarrow$ the second term(we are going to call it b) in the geometric progression is b=6 because a+ b= 31+b =37(now, we are going to call the terms in the geometric progression b/r, b and br where r is the ratio) $\Rightarrow$ a-d + b/r= 31-d + b/r = 31 $\Rightarrow$ d=b/r=6/r . Furthemore, a+d + br =51 ; 31 + 6/r +6r = 51 ; 6/r +6r = 20 $\Rightarrow$ r = 3 or 1/3 . We choose r=3 for seeing what happens. If r=3, d=6/3 =2 and the terms in the arithmetic progression would be 29, 31, 33 and the terms in the geometric progression would be 2, 6 and 18 and everything falls into place. And if r=1/3 , d=18 and everything falls into place too. The terms in the arithmetic progression would be 13, 31, 49 and the terms in the geometric progression would be 18, 6 and 2 so all the roads lead to Rome. 18+6+2=26

The sum of all items of both progessions is 31+37+51=119. Subtracting from this the sum of the arithmetic progession gives the sum of the geometric progression. So result is 119-93=26.