Question

I NEEED HELPPP!!!! 17+18+19+ … +42+43

Answer 1

Let

S = 1 + 2 + 3 + … + (n - 2) + (n - 1) + n

Reversing the order of terms, we have

S = n + (n - 1) + (n - 2) + … + 3 + 2 + 1

and we observe that terms in the same position on the right side add up to n + 1.

2S = (1 + n) + (2 + (n - 1)) + (3 + (n - 2)) + … + (n + 1)

There are n terms in the sum, so each the right side is simply n copies of n + 1, and

2S = n (n + 1)

S = n (n + 1)/2

Now,

17 + 18 + 19 + … + 41 + 42 + 43

= (1 + 2 + 3 + … + 41 + 42 + 43) - (1 + 2 + 3 + … + 14 + 15 + 16)

= (S with 43 terms) - (S with 16 terms)

= 43 (43 + 1)/2 - 16 (16 + 1)/2

= 810