Arithmetic Series and Means

Alivio, Barnachea, Garfin, Perez, and Pajares

Ms. Marissa De Ocampo

MEANS SERIES

a

It is the sum of an arithmetic sequence 𝑆𝑛 =

𝑛

2[2𝑎1 + 𝑛 − 1 𝑑]

Use if the last term is

not given

Arithmetic Series

Use only if the first and last term

are given

𝑆𝑛 =𝑛

2𝑎1 + 𝑎𝑛

Examples with first and last term formula

An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and

so on and has 30 rows of seats. How many seats are in the theatre?

There are 20 seats on the first row, 24 on the second row, and 28 on the third row. Each row has four

more seats than the one before it. Since we are adding four to each row, this is an arithmetic

sequence of numbers that we will be adding up. So we now know that our goal is to find an arithmetic

series. The formula for an arithmetic series is To solve this problem we need n, a1, and an. In this

problem, n will be equal to 30 because we are being asked to find how many seats are in all 30 rows.

Or to add up the seats in the 30 rows. The first term in the sequence, a1, is 20 because the problem tells

us that the first row has 20 seats. The only thing left to do is find an which will be a30. To find a30 we

need the formula for the sequence and then substitute n = 30. The formula for an arithmetic sequence

is We already know that is a1 = 20, n = 30, and the common difference, d, is 4. So now we have So we

now know that there are 136 seats on the 30th row. We can use this back in our formula for the

arithmetic series.

Examples with first and last term formula

Suppose you go to work for a company that pays one penny on the first day, 2 cents on the second

day, 4 cents on the third day and so on. If the daily wage keeps doubling, what will you total income

be for working 31 days?

This problem is geometric because the problem says that the salary from the previous day is doubled,

or multiplied by 2. When we are multiplying by the same number each time, this is a geometric

sequence. But what do we need to do with this geometric sequence? The problem wants to know

TOTAL income after 31 days. When dealing with total amounts, like in the previous example, we need

to add the terms in a sequence. In this case, since we will be adding terms in a geometric sequence,

we will be finding a geometric series. So we need the formula for a geometric series. We need to know

n, a1, and r. We are told r = 2 when the problem says doubling and n = 31 since that’s how many things

we need to add up. We also know the first term which is 0.01 (the decimal amount for one penny). This

should give us enough information to find the answer. Almost 21 1/2 million dollars total for 31 days

work. Not a bad job!

Examples using the long Formula

Find the sum of the first n terms of the arithmetic series 4+9+14+⋯ Also find the sum

of the first 17 terms. Solution:

Here a1=4 and d=9−4=5

If Sn is the sum of the first n terms, then

Sn=n2[2a1+(n−1)d]

=n2[2(4)+(n−1)(5)]=n2[8+5n−5]=n2(5n−3)

Sn=n(5n−3)2 ------ (1)

For the sum of first 17 terms, we put n=17 in (1), i.e.

S17=17[5(17)−3]2=17(85−3)2=17(82)2=697

Examples using the long Formula

Find n and sum of the numbers in

the following series 3 + 6 + 9 + 12

+ x?

Here a1=3, d=6-3=3, n=5 x= a1+(n-1)d =

3+(5-1)3 = 15

Sn=n/2*(2a1+(n-1)*d)

Sn=5/2*(2*3+(5-1)3)=5/2*18 = 45

Arithmetic Means -terms in between the

arithmetic extremes of a finite arithmetic

sequence

Term -any number in a sequence

Arithmetic Extremes -the first and last terms

of a finite arithmetic sequence

Arithmetic

Means Wherein: d= common

difference

an & ak =given

terms

n & k = position of

the given terms

𝑑 =𝑎𝑛 − 𝑎𝑘𝑛 − 𝑘

Formula of Difference

1. Insert four arithmetic means between 1/3

and 11/3. 1/3,_____,_____,_____,_______,11/3 Process:

GIVEN: a1=1/3 ; a6= 11/3

UNKNOWN: four arithmetic means

between 1/3 and 11/3

FORMULA TO BE USED: d=an- ak / n-k

SOLUTION: d=an- ak / n-k

d= (11/3)-(1/3)/ 6-1

d=(10/3)/5

d= 2/3

1/3 +2/3=1 1+ 2/3=5/3 5/3 +2/3=7/3

7/3+2/3=3 3+2/3=11/3

FINAL ANSWER: 1/3, 1, 5/3, 7/3, 3, 11/3

2. Insert three arithmetic means

between -16 and -8-16, ___, ___,

___, -8 Given: a1=-16, a5=-8

Unknown: three arithmetic means

between -16 and -8

Formula: d=an- ak / n-k

Solution: d=(-8)-(-16)/5-1

8/4

d=2

-16+2=-14 -14+2=-12 -12+2=-10 -

10+2=-8

Final Answer:

-16, -14, -12, -10, -8

References

http://planetmath.org/node/88129

http://www.regentsprep.org/regents/math/al

gtrig/atp2/arithseq.htm