Saturday, October 5, 2024

Sum of Sequential Integers (featuring Swiss Micros DM32)

 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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