## InfiniteGeometricProgression

Let us consider the G.P.

1, 2/3, 4/9, . . . .

The sum to n terms of this G.P. is: What happens if the number of terms n becomes larger and larger?

Let us study the behavior of (2/3)n when n becomes sufficiently large.

 n (2/3)n 1 0.6667 5 0.13168724280 10 0.1734152992 20 0.00030072866 40 0.00000009043772695

We see that as n becomes larger and larger, (2/3)n becomes smaller and smaller and approaches zero.

Mathematically, we say that as n becomes sufficiently large, (2/3)n approaches zero. (Also note that although (2/3)n approaches zero, it is never equal to zero). We further say that sn approaches 3 as n becomes sufficiently large.

We write as n becomes sufficiently large as: (2/3)n → 0 and sn → 3. or as (2/3)n tends to 0 then sn tends to 3

In the above example, we observe that r < 1. In fact, it can be proven that if r < 1, as n becomes large, rn 0.

We have a G.P. whose first term is “a” and common ratio is “r”. Since |r| < 1, rn → 0 and, →0

consequently,

Hence, sn = In other words, the sum of an infinite number of terms of a decreasing G.P. is a / (1 – r).

Briefly, we write the sum to infinity as a / (1 – r)

a
Thus, S ∞ =  ——  ( ∞ is the symbol for infinity)
1 - r

#### Examples

1)    Find the sum to infinity of the G.P. 5, 20/7, 80/49, . . .

Solution: 2)    Find S∞ for the G.P. -3/4, 3/16, -3/64, . . .

Solution: We now have a beautiful application of the sum to infinity of a G.P. with r < 1.

We know that rational numbers have a given non-terminating recurring decimal on expansion.

For example, 2/3 = 0.666 - - -

We will use the sum of an infinite G.P. to find the rational number of a given non-terminating recurring decimal.

For example, take the number 0.333. . .

We can write this as 0.3 + 0.03 + 0.003 + . . .

It this the sum of an infinite G.P. with a = 0.3 and r = 0.1 (r < 1)

What is its sum?

0.3         0.3
It is =  ———  =  —— = 1/3
1 – 0.1       0.9 #### Try these questions

I)    Find the following:

1. Find the sum to infinity of the G.P. . 1/2, 1/4, 1/8, 1/16, 1/32, . . .
2. Find a rational number, which when expressed as a decimal, will have as its expansion.
3. Find a rational number, which when expressed as a decimal will have as its expansion.
4. II)    Find S in the following G.P.

5. 1, 1/3, 1/9, . . .
6. 7, – 1, 1/7, – 1/49, . . .
7. 6, 1.2, 0.24, . . .
8. 50, 42.5, 36.125, . . .
9. 0.3, 0.18, 0.108, . . .
10. 10, – 9, 8.1, . . .
11. 3, 1/3, 1/9, . . .
12. III) For each of the following decimals, find a rational number, which will have as its expansion .

13. The first term of a G.P . is 2 and the sum to infinity is 6. Find the common ratio?
14. The common ratio of a G.P. . is – 4/5 and the sum to infinity is 80/9. Find the first term?
15. Find the sum of the series 1, 5, 25, . . .
16. Find the sum of 2/3, 1/3, 1/6, . . .
17. Find the sum of 1, 2/3, 4/9, . . . to infinity

#### Answers to Practice Problems

I)    Find the following

1. Find the sum to infinity of the G.P. . 1/2, 1/4, 1/8, 1/16, 1/32, . . .
Solution:
Here a = 1/2; r = 1/2 also r < 1
a
S =  ——
1 – r

1/2          1/2
S =  ———  =  ——  =  1
1 – 1/2       ½
2. Find a rational number, which when expressed as a decimal, will have as its expansion.
Solution:
We write = 0.234444 . . .
= 0.23 + 0.004 + 0.0004 + . . .
a
Here a = 0.004; r = 0.1 we’ll use S =   ——
1 – r
0.004
= 0.23 +  ———
1 – 0.1
0.23 + 0.004        0.23 + 4
=           ———  =          ——
0.9                  900
207 + 4
=  ————
900
= 211/900
Thus this is the required rational number.
3. Find a rational number, which when expressed as a decimal will have as its expansion
Solution:
We write = 1.56565656 . . .
= 1 + 0.56 + 0.0056 + . . .
0.56
= 1+  ————        (here a = 0.56; r = 0.01)
1 – 0.01
= 1 + 0.56/0.99
= 1 + 56/99
= 155/99.

4. II)    Find S in the following G.P. .
5. 1, 1/3, 1/9, . . .
Solution:
Here a = 1; r = 1/3
1              1
S =  ———  =  ——  =  1 3/2  =  3/2
1 – 1/3       2/3

6. 7, – 1, 1/7, – 1/49, . . .
Solution:
Here a = 7; r = – 1/7
7                   7              7
S =  —————  =  ————  =  ——
1 – (– 1/7)         1 + 1/7         8/7
=  7 7/8 = 49/8

7. 6, 1.2, 0.24, . . .
Solution:
Here a = 6; r = 0.2
6
S =  ———  =  6/0.8  =  60/8  =  7.5
1 – 0.2

8. 50, 42.5, 36.125, . . .
Solution:
Here a = 50; r = 0.85
50            50
S =  ———  =  ——  =  5000/15  =  333.3311:18 AM 9/18/2004–
1 – 0.85    0.15

9. 0.3, 0.18, 0.108, . . .
Solution:
Here a = 0.3; r = 0.6
0.3
S =  ———  =  0.3/0.4 =  3/4.
1 – 0.6

10. 10, – 9, 8.1, . . .
Solution:
Here a = 10; r = – 9/10
10                   10              10
S =  —————  =  ————  =  ———
1 – (– 9/10)        1 + 9/10       19/10

=  10 10/19

5
=   5 —–
9

11. 3, 1/3, 1/9, . . .
Solution:
Here a = 3:r = 1/9
3                       3
S = ------------- = ------------- = 3 9/8 = 27/8
1 - 1/9                   8/9

12. III) For each of the following decimals, find a rational number, which will have as its expansion
13. The first term of a G.P. . is 2 and the sum to infinity is 6. Find the common ratio?
Solution:
Here a = 2; S = 6; r = ?
a
S =  ——
1 – r

2
6 =  ——
1 – r
6(1 – r) = 2
6 – 6r – 2 = 0
4 – 6r = 0
6r = 4
r = 4/6 = 2/3

14. The common ratio of a G.P. . is – 4/5 and the sum to infinity is 80/9 find the first term?
Solution:
Here r = – 4/5; S = 80/9; a = ?
a
S =  ————
1 – (–4/5)

80          a
—–  =  ———
9       1 + 4/5
80          a
—–  =  ———
9          9/5
80/9 =  a 5/9
5a/9 =  80/9
5a  =  80
a = 80/5 = 16

15. Find the sum of the series 1, 5, 25, . . .
Solution:
Here a= 1; r = 5
S = 1 /(1– 5)
= 1/ –4
= –1/4>
16. Find the sum of 2/3, 1/3, 1/6, . . .
Solution:

2/3
S =  ———
1 – 1/2
2/3
=  ——  =  2/3 –2/1 = –4/3.
–1/2

17. Find the sum of 1, 2/3, 4/9, . . . to infinity
Solution:
1
S =  ———
(1 – 2/3)
1
=  ——  =  1 3/1 = 3.
1/3