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## Infinite Geometric Progressions

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, consequently, →0

 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

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

II) Find S in the following G.P.

 4 1, 1/3, 1/9, . . . 5 7, – 1, 1/7, – 1/49, . . . 6 6, 1.2, 0.24, . . . 7 50, 42.5, 36.125, . . . 8 0.3, 0.18, 0.108, . . . 9 10, – 9, 8.1, . . . 10 3, 1/3, 1/9, . . .

III) For each of the following decimals, find a rational number, which will have as its expansion .

 11 The first term of a G.P . is 2 and the sum to infinity is 6. Find the common ratio? 12 The common ratio of a G.P. . is – 4/5 and the sum to infinity is 80/9. Find the first term? 13 Find the sum of the series 1, 5, 25, . . . 14 Find the sum of 2/3, 1/3, 1/6, . . . 15 Find the sum of 1, 2/3, 4/9, . . . to infinity

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.

II) Find S in the following G.P. .

 4. 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 5. 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 6. 6, 1.2, 0.24, . . .      Solution: Here a = 6; r = 0.2            6 S =  ———  =  6/0.8  =  60/8  =  7.5          1 – 0.2 7. 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 8. 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 9. 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 10. 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

III) For each of the following decimals, find a rational number, which will have as its      expansion .

 11. 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 12. 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 13. Find the sum of the series 1, 5, 25, . . . Solution: Here a= 1; r = 5 S = 1 /(1– 5)      = 1/ –4      = –1/4 14. 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 15. Find the sum of 1, 2/3, 4/9, . . . to infinity Solution:             1 S =  ———         (1 – 2/3)            1      =  ——  =  1 3/1 = 3.           1/3