Which term of the Arithmetic Progression, AP: 3, 8, 13, 18 …… is 78?
$
  {\text{A}}{\text{. 12}} \\
  {\text{B}}{\text{. 16}} \\
  {\text{C}}{\text{. 10}} \\
  {\text{D}}{\text{. 18}} \\
$

Hint – To find the term which is 78, we apply the formula of the ${{\text{n}}^{{\text{th}}}}$term of a progression. For which the first term and the common difference is found from the given series.

Complete step-by-step answer:

Given data,
The progression is in AP: 3, 8, 13, 18 ……
Here the first term of the series a = 3
And the common difference d = 8 – 3 = 13 – 8 = 5.

Let the term at which the number 78 occurs be ‘n’
We know the ${{\text{n}}^{{\text{th}}}}$term of a progression is given by ${{\text{t}}_{\text{n}}} = {\text{a + }}\left( {{\text{n - 1}}} \right){\text{d}}$.
Here a = 3, d = 5 and ${{\text{t}}_{\text{n}}} = 78 = 3{\text{ + }}\left( {{\text{n - 1}}} \right)5$
$
   \Rightarrow 78 = 3{\text{ + 5n - 5}} \\
   \Rightarrow {\text{78 + 2 = 5n}} \\
   \Rightarrow {\text{n = }}\dfrac{{{\text{80}}}}{5} \\
   \Rightarrow {\text{n = 16}} \\
$
Hence Option B is the right answer.

Note – In order to solve this type of problems the key is to know the formulae of terms like the ${{\text{n}}^{{\text{th}}}}$term of an AP and in some cases the sum of n terms in an AP is also used. It is an important step to identify given 78 is the value of some term in the series. Also, sum of n terms in an AP is given by${{\text{S}}_{\text{n}}} = \dfrac{{\text{n}}}{2}\left( {{\text{2a + }}\left( {{\text{n - 1}}} \right){\text{d}}} \right)$.