Sum of Sequential Integers (featuring Swiss Micros DM32)
What is the sum of the following:
100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999?
It’s doable on a calculator, but going straight forward will require a lot of keystrokes (unless you have access to the sigma function (Σ)).
Note that:
100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999
= 100 + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)
= (100 + 0) + (100 + 1) + (100 + 2) + (100 + 3) + (100 + 4) + …. + (100 + 897) + (100 + 898) + (100 + 899)
Notice the sum starts with a base, 100. The sequence goes for 900 terms in the form of 100+n where n = 0 to 899. Let’s look at a general case.
Let x be a base integer, and S be the sum as:
S = (x + 0) + (x + 1) + (x + 2) + (x + 3) + …. + (x + n)
Rearranging the terms leads us to:
S = (x + x + x + x + … + x) + (0 + 1 + 2 + 3 + 4 + … + n)
There are n + 1 pairs:
S = (x * (n + 1)) + (0 + 1 + 2 + 3 + 4 + … + n)
S = (x * (n + 1)) + (1 + 2 + 3 + 4 + … + n)
The sum of Σ( k, k = 1 to k = n) = 1 + 2 + 3 + 4 + … + n = n * (n + 1) / 2
Then:
S = (x * (n + 1)) + n * (n + 1) / 2
Let’s look at some specific examples:
n = 1
S = (x + 0) + (x + 1)
S = (x + x) + (0 +1)
S = 2 * x + 1
n = 2
S = (x + 0) + (x + 1) + (x + 2)
S = (x + x + x) + (0 + 1 + 2)
S = 3 * x + 3
2 * 3 / 2 = 3
n = 3
S = (x + 0) + (x + 1) + (x + 2) + (x + 3)
S = (x + x + x + x) + (0 + 1 + 2 + 3)
S = 4 * x + 6
3 * 4 / 2 = 6
S = 500 + 501 +502 + 503
S = (500 + 0) + (500 + 1) + (500 + 2) + (500 + 3)
S = (500 + 500 + 500 + 500) + (0 + 1 + 2 + 3)
S = ((3 + 1) * 500) + (3 * 4 / 2)
S = 2000 + 6
S = 2006
S = 250 + 251 + 252 + 253 + … + 269 + 270
Base = 250
n = 270 – 250 = 20
Then:
S = 250 * (20 + 1) + 20 * 21 / 2
S = 5460
Let’s go our original problem:
S = 100 + 101 + 102 + 103 + 104 + 105 + … + 997 + 998 + 999
Base = 100
n = 999 -100 = 899
Then:
S = 100 * (899 + 1) + 899 * 900 / 2
S = 90000 + 404550
S = 494550
The program code calculates the sum:
Swiss Micros DM32 Program (also HP 32SII)
S01 LBL S
S02 INPUT X
S03 INPUT N
S04 RCL N
S05 1
S06 +
S07 RCL× N
S08 RCL N
S09 1
S10 RCL+ N
S11 ×
S12 2
S13 ÷
S14 +
S15 RTN
Eddie
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