XIV.

Ga = bG. G^2 = ab, lintand G = ± ab. Formula (2)

, the geometrical mean of any two numbers is the square root of their product.

177. To Find the Sum of Any Number of Terms of a Geometrical Progression.

If $l$ denote the last term, $a$ the first term, $n$ the number of terms, $r$ the common ratio, and $s$ the sum of the $n$ terms, then

s = a + ar + ar^2 + ar^3 + + ar^n - 1. lint Multiply by $r$, rs = ar + ar^2 + ar^3 + + ar^n - 1 + ar^n.

Therefore, by subtracting the first equation from the second,

rs - s = ar^n - a, lintor (r - 1)s = a(r^n - 1). s = a(r^n - 1)r - 1. Formula (3)

When $r$ is $<1$, this formula will be more convenient if written $s=\frac{a(1-r^n)}{1-r}.$

1. Find the sum of $8$ terms of the series $1,2,4,\dots .$

lint Here a = 1, r = 2, n = 8. lint From formula (3), s = 1(2^8 - 1) = 255.

2. Find the sum of $6$ terms of the series $2,3,\frac{9}{2},\dots .$

Exercise 77.

1. Find the 5th term of $3$, $9$, $27$, $\dots$.

2. Find the 7th term of $3$, $6$, $12$, $\dots$.

3. Find the 8th term of $6$, $3$, $\frac{3}{2}$, $\dots$.

4. Find the 9th term of $1$, $-2$, $4$, $\dots$.

5. Find the geometrical mean between $2$ and $8$.

6. Find the common ratio if the 1st and 3d terms are $2$ and $32$.

Find the sum of the series:

7. $3$, $9$, $27$, $\dots$ to $6$ terms.

8. $3$, $6$, $12$, $\dots$ to $8$ terms.

9. $6$, $3$, $\frac{3}{2}$, $\dots$ to $7$ terms.

10. $8$, $4$, $2$, $\dots$ to $8$ terms.

11. $64$, $32$, $16$, $\dots$ to $9$ terms.

12. $64$, $-32$, $16$, $\dots$ to $5$ terms.