Arithmetic Series

19 May 2011

Summations Summation – the sum of the terms in a

sequence

{2, 4, 6, 8} → 2 + 4 + 6 + 8 = 20

Represented by a capital Sigma

Summation Notation

k

1nnuSigma

(Summation Symbol)

Upper Bound (Ending Term #)

Lower Bound (Starting Term #)

Sequence

Example #1

4

1nn2

Example #2

3

1n)3n(

Example #3

)2n3(3

1n

Your Turn: Find the sum:

5

1n)7n3(

4

1n)n45(

Your Turn: Find the sum:

5

1n)n37(

4

1n]4)1n(3[

Your Turn: Find the sum:

5

1n

2 )n30(

4

1n)2n(n

Partial Sums of Arithmetic Sequences – Formula #1

Good to use when you know the 1st term AND the last term

k

1nk1n )uu(

2

ku

# of terms

1st term last term

Formula #1 – Example #1Find the partial sum:

k = 9, u1 = 6, u9 = –24

Formula #1 – Example #2Find the partial sum:

k = 6, u1 = – 4, u6 = 14

Formula #1 – Example #3Find the partial sum:

k = 10, u1 = 0, u10 = 30

Your Turn:

Find the partial sum:

1. k = 8, u1 = 7, u8 = 42

2. k = 5, u1 = –21, u5 = 11

3. k = 6, u1 = 16, u6 = –19

Partial Sums of Arithmetic Sequences – Formula #2

Good to use when you know the 1st term, the # of terms AND the common difference

k

1n1n d

2

)1k(kkuu

# of terms

1st term common difference

Formula #2 – Example #1Find the partial sum:

k = 12, u1 = –8, d = 5

Formula #2 – Example #2Find the partial sum:

k = 6, u1 = 2, d = 5

Formula #2 – Example #3Find the partial sum:

k = 7, u1 = ¾, d = –½

Your Turn:

Find the partial sum:

1. k = 4, u1 = 39, d = 10

2. k = 5, u1 = 22, d = 6

3. k = 7, u1 = 6, d = 5

Choosing the Right Partial Sum Formula

Do you have the last term or the constant difference?

k

1n1n d

2

)1k(kkuu

k

1nk1n )uu(

2

ku

Examples Identify the correct partial sum formula:

1. k = 6, u1 = 10, d = –3

2. k = 12, u1 = 4, u12 = 100

Your Turn: Identify the correct partial sum formula

and solve for the partial sum

1. k = 11, u1 = 10, d = 2

2. k = 10, u1 = 4, u10 = 22

3. k = 16, u1 = 20, d = 7

4. k = 15, u1 = 20, d = 10

5. k = 13, u1 = –18, u13 = –102