If the 4th term of an a.p is 27 and the 7th term is 243.find the common ratio and first term
We know that the nth term of an arithmetic progression (AP) can be found using the formula:
\[a_n = a + (n-1)d\]
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In this case, we are given that the 4th term is 27, so we can substitute these values into the formula:
\[27 = a + (4-1)d\]
Simplifying this equation, we get:
\[27 = a + 3d \quad \text{(equation 1)}\]
We are also given that the 7th term is 243, so we can substitute these values into the formula:
\[243 = a + (7-1)d\]
Simplifying this equation, we get:
\[243 = a + 6d \quad \text{(equation 2)}\]
Now, we can solve these two equations to find the common difference (\(d\)) and the first term (\(a\)).
First, let's subtract equation 1 from equation 2 to eliminate \(a\):
\begin{align*}
(243 - 27) &= (a + 6d) - (a + 3d) \\
216 &= 3d \\
d &= \frac{216}{3} \\
d &= 72
\end{align*}
Now, substitute the value of \(d\) into equation 1 to find \(a\):
\begin{align*}
27 &= a + 3(72) \\
27 &= a + 216 \\
a &= 27 - 216 \\
a &= -189
\end{align*}
Therefore, the common ratio (\(d\)) is 72, and the first term (\(a\)) is -189.