# MATH EXPONENTIAL HELP?

2. How many terms are there in the sequence: 10, -6, 18/5, -54/25, .... -4374/15625 ?

Relevance

So this is geometric series

-6/10  = -3/5

18/5 / -6   = -18/30 =-3/5

so r = -  3/5

a(1) = 10

so standard formula is

a(n) = a(1)*r^(n-1)

a(n) =  10*(-3/5)^(n-1)

so the last term is

-4374/15625  = 10*(-3/5)^(n-1)

since it is negative

n-1 = odd

so n = even

This allow to remove the -1

-4374/15625 = 10*(-1)^(odd n-1) (3/5)^(n-1)

-4374/15625 = -10* (3/5)^(n-1)

divide both sides by -10

4374/156250 = (3/5)^(n-1)

take ln of both sides

ln(4374/165250) =  (n-1)*ln(3/5)

(n-1) =   ln(4374/156250)/ ln(3/5)

n-1 = 7

n = 7+1 = 8

n = 8

• How many terms are there in the sequence:

10, -6, 18/5, -54/25, .... -4374/15625

a_n = -2 (-1)^n 3^(n - 1) 5^(2 - n)

-4374/15625 = -2 (-1)^n 3^(n - 1) 5^(2 - n)

Integer solution:

n = 8

• Work at it a little longer and try to find a pattern.

Here, convince yourself that this sequence can be written as

(-1)^0 * 2* 3^0 / 5^(-1), (-1)^1 * 2* 3^1 / 5^0, (-1)^2 * 2* 3² / 5^1, (-1)^3 * 2 * 3^3 / 5² ...

See the pattern?

You can also simplify this a little if need be. Write it with the parameter "n".

Ok...   -2 * 25 / 3 * (-3/5)ⁿ

If n=1 you get? If n=2 you get? Does it generate your sequence?

So, how many terms are there ?