A.P. Series Common Difference Calculation

We are given the details of an Arithmetic Progression (A.P. series):

• The $5^{\text{th}}$ term ($a_5$) is 52.
• The $17^{\text{th}}$ term ($a_{17}$) is 184.

We need to find the common difference ($d$) of this A.P. series.

Applying the A.P. Formula

The formula for the n-th term of an A.P. is:

$a_n = a_1 + (n-1)d$

Where $a_1$ is the first term and $d$ is the common difference.

Using this formula, we can set up two equations based on the given information:

1. For the $5^{\text{th}}$ term ($n=5$):

   $a_5 = a_1 + (5-1)d$

   $52 = a_1 + 4d \quad \text{(Equation 1)}$

2. For the $17^{\text{th}}$ term ($n=17$):

   $a_{17} = a_1 + (17-1)d$

   $184 = a_1 + 16d \quad \text{(Equation 2)}$

Solving for the Common Difference

To find the common difference ($d$), we can subtract Equation 1 from Equation 2:

$ (a_1 + 16d) - (a_1 + 4d) = 184 - 52 $

Simplify the equation:

$ a_1 + 16d - a_1 - 4d = 132 $

$ 12d = 132 $

Now, isolate $d$ by dividing both sides by 12:

$ d = \frac{132}{12} $

$ d = 11 $

Therefore, the common difference of the A.P. series is 11.