Page 60 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

means. Let’s practice. What does

10

5k
k=8

mean? First, what are we adding? This time, we are adding 5 times k . Where do we start? We

start with k =8. When do we stop? We stop with k =10. So,

10

5k =5 ∗ 8+ 5 ∗ 9+ 5 ∗ 10 =40+ 45 +50= 135.
k=8

What is

5
2
j ?
j=2

Well, this time I called it j instead of k , just to emphasize that there isn’t anything special
about using the letter k. What matters is the meaning. Here,

5
2
j
j=2

means we sum the integers squared. Where do we start? We start with 2. Where do we stop?
We stop at 5. So,

5

2
2
2
2
2
j =2 +3 +4 +5 =4 +9+16+ 25 =54.
j=2
Play with summation notation until you feel comfortable with it. Then prove by induction the
following formulas

n
2 n(n +1)(2n +1)
k = ,
6
k=1
n 2 2
3 n (n +1)
k = .
4
k=1

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