To find x that turns the sequence -1, 5, and 2 into a geometric sequence when added to each term, we set up the equation for the ratios of terms and solve for x, resulting in x = -3.
To solve this problem, we must understand that in a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the ratio (r). The given terms are -1, 5, and 2; we will denote these as a1, a2, and a3 respectively. Our goal is to find the value of x that, when added to each term, results in a geometric sequence.
Let's think about the terms as a1 + x, a2 + x, and a3 + x. For the sequence to be geometric, the ratio between (a2 + x) / (a1 + x) must equal the ratio (a3 + x) / (a2 + x). Setting these ratios equal to each other gives us the equation:
(5 + x) / (-1 + x) = (2 + x) / (5 + x)
Cross-multiplication yields:
(5 + x)(5 + x) = (-1 + x)(2 + x)
Expanding both sides gives us:
25 + 10x + x² = -2 - x + 2x + x²
Combining like terms and rearranging the equation, we can solve for x:
25 + 10x = -2 + x
9x = -27
x = -3
Therefore, the value of x that needs to be added to each term is -3.
The probable question may be:
In a geometric sequence, the terms are -1, 5, and 2. If a certain number x is added to each term, what is the value of x?
Additional Information:
Imagine you have a sequence of numbers, and each number is like a building block in a pattern. The given sequence -1, 5, 2 is like a secret code, and we want to discover the hidden number x that, when added to each term, unlocks the pattern.
Let's call the terms in the sequence a1, a2, and a3. So, we have:
a1=−1
a2=5
a3=2
Now, we're looking for xx to transform each term:
a1+x, a2+x, a3+x