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

1 answer

To find the common ratio and the first term of the arithmetic progression (AP), we can use the formula for the n-th term of an AP:

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

Given that the second term of the AP is 12:

\[a_2 = a_1 + (2-1)d = a_1 + d = 12\]

Given that the fifth term of the AP is 96:

\[a_5 = a_1 + (5-1)d = a_1 + 4d = 96\]

Now we have two equations:

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

Subtracting equation (1) from equation (2):

\[3d = 84\]

\[d = 28\]

Plugging the value of \(d\) back into equation (1):

\[a_1 + 28 = 12\]

\[a_1 = -16\]

Therefore, the common ratio is 28 and the first term of the AP is -16.