Sigma notation - Mathcentre

Suppose we have the sum of a constant times k. What does this give us? For example,
4∑
3k = (3 × 1) + (3 × 2) + (3 × 3) + (3 × 4)
k=1
= 3 × (1 + 2 + 3 + 4)
= 3 × 10
= 30 .
But we can see from this calculation that the result also equals
4∑
3 × (1 + 2 + 3 + 4) = 3 k ,
so that
4∑
3k = 3
k=1
4∑
k .
In general, we can say that
n∑
ck = (c × 1) + (c × 2) + . . . + (c × n)
k=1
k=1
= c × (1 + ... + n)
n∑
= c k .
k=1
Suppose we have the sum of k plus a constant. What does this give us? For example,
4∑
(k + 2) = (1 + 2) + (2 + 2) + (3 + 2) + (4 + 2)
k=1
k=1
= (1 + 2 + 3 + 4) + (4 × 2)
= 10 + 8
= 18 .
But we can see from this calculation that the result also equals
so that
(4 × 2) + (1 + 2 + 3 + 4) = (4 × 2) +
4∑
(k + 2) = (4 × 2) +
k=1
4∑
k .
k=1
4∑
k ,
In general, we can say that
n∑
(k + c) = (1 + c) + (2 + c) + . . . + (n + c)
k=1
= (c + c + . . . + c)
} {{ }
n times
n∑
= nc + k .
k=1
k=1
+ (1 + 2 + . . . + n)
7 c○ mathcentre July 18, 2005