Question

The nth term of an A.P. is \(3n + 2\). The common difference is :

• A. 8
• B. 2
• C. 5
• D. 3

Hint:

For any Arithmetic Progression whose \(n\)-th term is represented as a linear expression \(a_n = An + B\), the common difference is always equal to the coefficient of \(n\) (which is \(A\)).
In this case, since \(a_n = 3n + 2\), the coefficient of \(n\) is 3, so the common difference is immediately 3. This saves you from performing any calculation!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The \(n\)-th term of an Arithmetic Progression (A.P.) is given as \(a_n = 3n + 2\). We need to find the common difference of this A.P.

Step 2: Key Formula or Approach:
The common difference \(d\) of an Arithmetic Progression is the difference between any term and its preceding term. Mathematically, it is expressed as:
\[ d = a_{n} - a_{n-1} \]
Alternatively, we can find the first two consecutive terms \(a_1\) and \(a_2\) by substituting \(n=1\) and \(n=2\), and then find \(d = a_2 - a_1\).

Step 3: Detailed Explanation:

Method 1: Finding consecutive terms
1. Calculate the first term \(a_1\) by substituting \(n = 1\):
\[ a_1 = 3(1) + 2 = 3 + 2 = 5 \]
2. Calculate the second term \(a_2\) by substituting \(n = 2\):
\[ a_2 = 3(2) + 2 = 6 + 2 = 8 \]
3. The common difference \(d\) is:
\[ d = a_2 - a_1 = 8 - 5 = 3 \]


Method 2: General Algebraic Method
1. The \(n\)-th term is \(a_n = 3n + 2\).
2. The \((n-1)\)-th term is:
\[ a_{n-1} = 3(n-1) + 2 = 3n - 3 + 2 = 3n - 1 \]
3. The common difference is:
\[ d = a_n - a_{n-1} = (3n + 2) - (3n - 1) \]
\[ d = 3n + 2 - 3n + 1 = 3 \]
Both methods consistently yield a common difference of 3.

Step 4: Final Answer:
The common difference of the A.P. is 3, which corresponds to option (D).