LESSON 7-4 : Arithmetic Series

Terminology

·  Series – the sum of the terms of a .

·  Arithmetic Series - the sum of the terms of an sequence.

·  Sn - the partial sum of the first n terms of a sequence where,

Sn = t1 + t2 + t3 + … + tn-1 + tn

·  The arithmetic series is:

Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + [a + d(n – 2)] + [a + d(n – 1)]

The General Formula for Sn

How did Karl Friedrich Gauss find the sum of the first 100 natural numbers?

Solution: S100 = 1 + 2 + 3 + . . . + 99 + 100 ... (1)

write in reverse: S100 = 100 + 99 + 98 + . . . + 2 + 1 ...(2)

Add (1) + (2) : 2 S100= 101 + 101 + 101 +. . . + 101 + 101

2 S100= 100(101)

2 2

S100 = 50(101)

S100 = 5050

How can you find the sum of any number of terms in an arithmetic sequence?

Solution: Sn = a + (a + d) + (a + 2d) + . . . + (tn - d) + tn ...(1)

write in reverse: Sn = tn + (tn – d) + (tn – 2d) + . . . + (a + d) + a ...(2)

Add (1) + (2) : 2 Sn = (a + tn) + (a + tn)+ (a + tn)+ . . . + (a + tn) + (a + tn)

2 Sn= n(a + tn)

2 2

Sn = n(a + tn)

2

or Sn = n(t1 + tn) Arithmetic Series given last term, tn

2

Also, since tn = a + d(n-1), Sn = n(a + tn)

2

Substitute for tn : Sn = n[a + a + d(n-1)]

2

Sn = n[2a + d(n-1)] Arithmetic Series given

2 common difference,

Examples:

1. For the series 2 + 11 + 20 + 29 + … , find S20.

2. The fifth term of an arithmetic series is 9, and the sum of the first 16 terms is 480. Find

the first three terms of the series.

t5 = 9

S16 =480

(1)  9 = a +(5 – 1)d à 9 = a + 4d

(2)  480 = 16/2(2a + (16 – 1)d) à 480 = 2a + 15d

When solving this system we get:

d = 6

a = -15

.: -15, -9, -3

3. Find the sum of the arithmetic series if the series is 3+7+11+…+479.

a = 3

d = 4

479 = 3 + (n – 1)4

479 = 3 – 4 + 4n

479 = 4n – 1

n = 120

4. A stadium has 25 rows. The 1st row has 40 seats. Each successive row has 1 more

seat. How many seats are there in the stadium?

5. Sukaina deposited $128 in her account. Each week, she deposits $7 less than the

previous week until she makes her last deposit of $9. Find the total value of her

deposits.

1. Find n: / 2. Find S18

The total value of her deposits is $1233.

Homefun: p. 452 #2, 3, 5, 6bdf, 10, 11, 13, 15