evaluate the following numbers

Question

50-53+49-52+48-51+...+0-3

Answer 1

If the intent is to find the sum of that set of numbers, then think of this set as two separate series:

0+1+2+....50

and

-53 + (-52) + (-51) + ….+(-3)

These are both arithmetic series with a common difference of 1. One way to find the sum of an arithmetic series is: (number of terms) times the (average of 1st and last terms):

S₅₁=51 (0+50)/2 (for the 1st series)

Find the sum of the 2nd series in the same way, then combine the results.

This Desmos graph shows some other ways to find the sum(s).

desmos.com/calculator/u9vdpsixva

Answer 2

alternating signs

in absolute values:

up 3, down 4, up 3, down 4

50, -53, 49, -52, 48, - 51, 47, -50, 46, -49, 45, -48, 44, -47, 43, ...0, -3

add the the 1st and every 3rd, 5th, and odd terms

is the sum of an arithmetic sequence with a1 =50, an = a1+ d(n-1) = 50-1(n-1) = 50-50n

sn = (n/2)(a1 + an)'= (51/2)(50+0) = 25(51=1275

do the same with -53,-52, ...-3 with d =+1

an =-53+(n-1) = 53n-53

sn = (n/2)(a1+an)

s51 = (51/2)(-53-3) =-56(51/2) = -28(51) =-1428

add the two sums together

25(51) -28(51) =-3(51) -153

50 to 0 has 51 terms

-53 to -3 has 51 terms

51 pairs where 2nd term is -3 less than 1st term

51 x -3 = -153

is a more simple way to calculate the sum