hi i need help for a math question. The question is about geometric sequence, it asking about " in 2004, the turbines generated 326 MW of wind energy, and it is projected that the amount will be 10,000 MW per year by 2010. If this growth were modeled by geometric sequence, determined the value of the annual growth rate from 2004 to 2010."

so far how i did it is:
2004-2010= 6 year difference then
10,000/6 =1666.666 and then 1666.666/326= 5.11
But the answer at the back of my text book say "1.77" i don't know where i am going wrong.. please help

let the rate of growth be r

then 326(1+r)^6 = 10,000
(1+r)^6 = 30.6748.. ( I stored it in memory for accuracy)
take 6th root of both sides
1+r = 30.6748^(1/6) = 1.76928...

or , using GS

a = 326
r = ?

2004 ---> term 1 = a
2010 ----> term 7 = ar^6
326 r^6 = 10000
r^6 = 10000/326 = 30.6748..
same as above
r = 30.6748..^(1/6) = 1.76928... <---- growth factor

When you divide by 10000/6 that's would be the avg rate of climb per year. Your not looking for that. You need to find the geometric sequence for the number 326 to 10k with a annual growth rate of 6 years. You do this by taking 1.77,your given answer, and multiplying 326 by 1.77 6 times. 326x1.77^6 which will roughly equal out to 10k MW.

To find the annual growth rate in a geometric sequence, we need to find the common ratio between consecutive terms.

In this case, we know that the turbines generated 326 MW in 2004 and are projected to generate 10,000 MW in 2010.

First, let's calculate the total growth over the 6-year period:

Total Growth = Final Value - Initial Value
Total Growth = 10,000 MW -326 MW
Total Growth = 9,674 MW

Next, we need to find the common ratio, which represents the rate of change between each year. Let's call this ratio "r."

We know that the formula for the nth term in a geometric sequence is given by:

An = A1 * r^(n-1)

Where:
An = nth term
A1 = first term
r = common ratio
n = number of terms

Since we are comparing the growth from 2004 to 2010, we can consider 2004 as the first term and 2010 as the last term (we will ignore any progress beyond 2010 for now).

So, using the formula:

10,000 = 326 * r^(6-1)
10,000/326 = r^5
30.67 = r^5

To find the value of r, we need to take the fifth root of both sides:

r = (30.67)^(1/5)
r = 1.7666 (rounded to four decimal places)

Therefore, the annual growth rate from 2004 to 2010 is approximately 1.7666 or 1.77 (rounded to two decimal places).

So your answer of 5.11 seems incorrect, and the correct answer is indeed 1.77.