The 8th term of a gp is _7/32. find its common ratio if its first term is 28.
a1 = 28
a8 = a1 ∙ r ⁷
7 / 32 = 28 ∙ r ⁷
Divide both sides by 28
( 7 / 32 ) / 28 = r ⁷
7 / 32 ∙ 28 = r ⁷
7 / 32 ∙ 7 ∙ 4 = r ⁷
1 / 32 ∙ 4 = r ⁷
1 / 128 = r ⁷
r = ⁷√ ( 1 / 128 )
r = ⁷√ 1 / ⁷√ 128
r = 1 / 2
I need answer
28
To find the common ratio of a geometric progression (GP) given the first term and the eighth term, we can use the formula:
\[T_n = a \cdot r^{(n-1)}\]
Where:
- \(T_n\) is the n-th term of the GP
- \(a\) is the first term of the GP
- \(r\) is the common ratio of the GP
- \(n\) is the term number
In this case, we are given the following information:
- The 8th term of the GP is \(-\frac{7}{32}\)
- The first term of the GP is 28
Substituting these values into the formula, we have:
\[-\frac{7}{32} = 28 \cdot r^{(8-1)}\]
Simplifying this equation, we get:
\[-\frac{7}{32} = 28 \cdot r^7\]
To find the common ratio, we need to isolate \(r\) on one side of the equation. We can start by dividing both sides of the equation by 28:
\[\frac{-7}{32 \cdot 28} = r^7\]
Now, we can take the seventh root of both sides to isolate \(r\):
\[\sqrt[7]{\frac{-7}{32 \cdot 28}} = r\]
Therefore, the common ratio of the geometric progression is \(\sqrt[7]{\frac{-7}{32 \cdot 28}}\) or approximately -0.5.