What number must be added to each of the numbers 0, 8, and 32 so that they form consecutive terms of a geometric sequence?
I don't understand what the question is asking first of all.
The answer is 4.
Help is much appreciated.
"A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1, 1/3,... are geometric, since you multiply by 2 and divide by 3, respectively, at each step."
-- http://www.purplemath.com/modules/series3.htm
A geometric sequence cannot start with 0, or all the terms will just stay 0.
So, you want n such that each term is a constant multiple of the one before.
(8+n)/(0+n) = (32+n)/(8+n)
(8+n)^2 = n(32+n)
64 + 16n + n^2 = 32n + n^2
64 = 16n
n=4
So, the sequence starts out 4,12,36,... with each term 3x the previous one.
The question is asking for a number that, when added to each of the given numbers (0, 8, and 32), will make them form consecutive terms of a geometric sequence.
A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant factor. In this case, the constant factor is called the common ratio.
To determine the common ratio, we can find the ratio between any two consecutive terms in the sequence. Let's take the ratio between the second term (8) and the first term (0): 8/0 = undefined (division by zero).
Since the common ratio cannot be determined using the given terms, we need to find the common ratio from another set of consecutive terms.
Let's consider the difference between the second and third terms: 32 - 8 = 24.
To find the common ratio, we can divide the third term (32) by the second term (8): 32/8 = 4.
Now that we know the common ratio is 4, we want to find the number that, when added to each of the given numbers, will make them form consecutive terms with a common ratio of 4.
For the first term (0), we need to find the next term by multiplying it by the common ratio: 0 * 4 = 0. Since 0 multiplied by any number is still 0, we can see that the first term is already in the correct position.
For the second term (8), we want to find the next term by multiplying it by the common ratio: 8 * 4 = 32.
Now we can observe that the second term (8) is correctly positioned between the first term (0) and the third term (32) to form a geometric sequence with a common ratio of 4.
Therefore, the number that needs to be added to each of the given numbers (0, 8, and 32) to form consecutive terms of a geometric sequence is 4.