the 2nd and 5th terms of a go are -7 and 56 respectively. find the
common ratio
first term
sum of the first five terms?
ar^4/ar = r^3 = 56/-7 = -8
So, now you have a and r, so you just need
5(r^5-1)/(r-1)
wrong
wrong sir
To find the common ratio of the geometric progression, we can use the formula:
\(a_n = a_1 \cdot r^{(n-1)}\)
Given that the 2nd term (\(a_2\)) is -7 and the 5th term (\(a_5\)) is 56, we can substitute those values into the formula to get:
\(a_2 = a_1 \cdot r^{(2-1)}\), which gives us:
-7 = \(a_1 \cdot r\)
\(a_5 = a_1 \cdot r^{(5-1)}\), which gives us:
56 = \(a_1 \cdot r^{4}\)
Now, we can solve these two equations simultaneously to find the values of \(a_1\) and \(r\).
First, divide the second equation by the first equation to eliminate \(a_1\):
\(\frac{56}{-7} = \frac{a_1 \cdot r^{4}}{a_1 \cdot r}\)
Simplifying:
\(-8 = r^3\)
Now, we can take the cube root of both sides to find the value of \(r\):
\(r = \sqrt[3]{-8}\)
\(r = -2\)
Now that we have the value of \(r\), we can substitute it into either of the original equations to find the value of \(a_1\). Let's use the equation \(a_2 = a_1 \cdot r\) and substitute -7 for \(a_2\) and -2 for \(r\):
-7 = \(a_1 \cdot (-2)\)
Solving for \(a_1\):
\(a_1 = \frac{-7}{-2}\)
\(a_1 = 3.5\)
So, the common ratio (\(r\)) is -2 and the first term (\(a_1\)) is 3.5.
To find the sum of the first five terms, we can use the formula for the sum of a geometric series:
\(S_n = \frac{a_1 \cdot (1 - r^n)}{1 - r}\)
Substituting the values of \(a_1 = 3.5\), \(r = -2\), and \(n = 5\):
\(S_5 = \frac{3.5 \cdot (1 - (-2)^5)}{1 - (-2)}\)
Simplifying:
\(S_5 = \frac{3.5 \cdot (1 - 32)}{1 + 2}\)
\(S_5 = \frac{3.5 \cdot (-31)}{3}\)
\(S_5 = \frac{-108.5}{3}\)
\(S_5 = -36.17\)
Therefore, the sum of the first five terms is approximately -36.17.