Precalc help

Question

Find identified term of the geometric sequence.

7th term: 8/5, -16/25, 32/125,-64/625,....

Answer 1

The formula for a geometric sequence is

 

a_n = a₁(r)^n-1, where r is the common ratio and n is the number of terms.

The common ratio is -2/5

 

a₇ = (8/5)(-2/5)^7-1

a₇ = (8/5)(-2/5)⁶

a₇ = (8/5)(64/15625)

a₇ =  512/78125

Answer 2

1 - 8 gets multiplied by 2 with every term

2 - every other term (even terms,  2nd, 4th, ...) gets multiplied by (-1)

3 - 5 gets multiplied by 5 for every term

4 - thus, the denominator would have to have five to the power of whatever the term is, in this case 7. 5^7

5 - the power of 2 at the numerator is 1 less than of the term. In this case - 7th term it is 6. 2^6

 

[i^n*8*2^(n-1)]/5^n

 

n starts at 1

 

i = square root of -1

Answer 3

Just a quick answer - the trick is to look at powers of the numbers in question.
(-1) raised to an even power makes a positive, to an odd power makes a negative.

Otherwise, we have powers of 2 in the numerator and of 5 in the denominator, which means negative powers

(-1)^(n+1) * 2^(n+2) * 5^(-n)

Play around with this, plug in numbers in place of n, and see if you get these results.  Remember, 5^-2 = 1/25.