The second term of Ap is 12 and the fifth term is 96 find the common ratio and the first term

Answer 1

To find the common ratio of an arithmetic progression (AP), we can use the formula for the nth term of an AP:

\[ a_n = a_1 + (n-1)d \]

Where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the position of the term, and \( d \) is the common difference.

Given that the second term is 12 and the fifth term is 96, we can form two equations:

For the second term (n=2):
\[ 12 = a_1 + (2-1)d \]
\[ 12 = a_1 + d \]

For the fifth term (n=5):
\[ 96 = a_1 + (5-1)d \]
\[ 96 = a_1 + 4d \]

Now, we have two simultaneous equations:

\[ 12 = a_1 + d \ \ \ \ \ (1) \]
\[ 96 = a_1 + 4d \ \ \ \ \ (2) \]

Subtracting equation (1) from equation (2) to eliminate \( a_1 \):
\[ 96 - 12 = 4d - d \]
\[ 84 = 3d \]
\[ d = 28 \]

Now substituting the value of \( d \) back into equation (1):
\[ 12 = a_1 + 28 \]
\[ a_1 = -16 \]

Therefore, the common ratio is \( d = 28 \) and the first term is \( a_1 = -16 \).