Arithmetic and Geometric Series (11.5)

Short cuts

POD

Consider a list of the first 100 positive integers.

What sort of sequence is it?

Add them. I’ll time how long it takes.

What do we call a sum of terms in a sequence?

POD

Consider a list of the first 100 positive integers.

It’s an arithmetic sequence, with d = 1.

We’re finding a series. We use sigma notation, or we can use Sn notation.

Want a short cut? It’s based off of something we did yesterday.

Arithmetic partial sums

To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

Arithmetic partial sums

To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 =

What do we do next? Can you think of a pattern?

Arithmetic partial sums

To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 =

11+11+11+11+11+11+11+11+11+11=110

We added the first and last terms.

Arithmetic partial sums

To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 =

11+11+11+11+11+11+11+11+11+11=110

We need only half of the number of sums here.

Arithmetic partial sums

To figure this out, start smaller. Add the first 10 positive integers. Remember what we did yesterday?

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 =

11+11+11+11+11+11+11+11+11+11=110

nn ttn

S 12

Arithmetic partial sums

Use this to answer the POD.

nn ttn

S 12

Arithmetic partial sums

Use this to answer the POD.

S100 = 100/2(1 + 100)

= 50(101)

= 5050

This problem was posed to Karl Friedrich Gauss (1777-1855) in third grade, and he determined the pattern.

Arithmetic partial sums

We have another tool, too. What is the formula for the nth term of an arithmetic sequence? How could we use it to change this formula?

nn ttn

S 12

Arithmetic partial sums

We have another tool, too. What is the formula for the nth term of an arithmetic sequence? How could we use it?

dntn

S

dntt

ttn

S

n

n

nn

)1(22

)1(

2

1

1

1

Arithmetic partial sums

We decide which tool to use based on the information given. What information would you need to either one?

dntn

S

ttn

S

n

nn

)1(22

2

1

1

Use them

1. Find the sum of the first 50 positive even integers. Which formula would you use?

How does this compare to the sum of the first 100 positive integers?

Use them

1. Find the sum of the first 50 positive even integers. Which formula would you use?

S50 = 50/2(2 + 100) = 25(102) = 2550

How does this compare to the sum of the first 100 positive integers?

Use them

1. Find the sum of the first 50 positive even integers.

Let’s look at it in summation notation.

2k 2 4 6 8 ...100k1

50

Use them

2. Find S12 for the series with t1 = 3 and d = -4

Use them

2. Find S12 for the series with t1 = 3 and d = -4.

S12 = 12/2(2(3) + (11)(-4))

= 6(6-44)

= 6(-38)

= -228

Geometric shortcut

The shortcut for a geometric series uses the first term and r.

r

rtS

n

n 1

11

Use it

3. Find S8 for 1 + 2 + 4 +8 +….

r

rtS

n

n 1

11

Use it

3. Find S8 for 1 + 2 + 4 +8 +….

r

rtS

n

n 1

11

2551

2561

21

211

8

8

S

Use it

4. Find S20 for the series with first term of 11 and

r = 1.3.

Use it

4. Find S20 for the series with first term of 11 and

r = 1.3.

820.6931)165.630(113.

050.18911

3.11

3.1111

20

20

S