Sunday, April 5, 2020

The Sum of a Constant

The Sum of a Constant

Introduction

What is the sum of the series:

∑ a from x= 0 to n   (a is a real or complex constant, n is a positive integer)

I may not be what you think.   Take a close look at the limits:  lower limit of 0, upper limit of n.   Assume the increment of x is 1. 

The sum of the series is (n + 1) * a.

Proof

Base case.   Let n = 1.  Then:

∑ a from x = 0 to 1

=  a + a   

= 2 * a

= (1 + 1) * a

The value a is added for the x=0 term.   The value a is added for the x=1 term.

Induction.   Assume for a positive integer k,  the series holds.  Then for the sum from x = 0 to x = k + 1:

∑ a from x = 0 to k+1

= ( ∑ a from x = 0 to k ) + ( ∑ a from x = k+1 to k+1 )

= (k + 1) * a + a

= k * a + a + a

= k * a + 2 * a

= (k + 2) * a    QED

Examples


Example 1:

∑ a from x = 0 to 2

= a + a + a

= 3 * a


Example 2:

∑ 6 from x = 0 to 11

= (11 + 1) * 6

= 12 * 6

= 72


Eddie

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