Question

In an A.P., the first term is 4 and the last term is 31. If sum of all the terms is 175, find the number of terms and the common difference.

Hint:

Using the sum formula \( S_n = \frac{n}{2}(a+l) \) is much faster than using \( S_n = \frac{n}{2}[2a + (n-1)d] \) when the last term is explicitly given.
It allows you to find \( n \) directly without dealing with quadratic equations or simultaneous equations.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Arithmetic Progressions (AP).
We are given the first term \( a \), the last term \( l \) (or \( a_n \)), and the sum of all terms \( S_n \) of an AP.
We need to calculate the total number of terms \( n \) and the common difference \( d \).

Step 2: Key Formula or Approach:
- The sum of \( n \) terms of an AP when the first and last terms are known is:
\[ S_n = \frac{n}{2}(a + l) \]
- The general term (or last term) of an AP is given by:
\[ l = a + (n - 1)d \]

Step 3: Detailed Explanation:
1. Note down the given parameters from the question:
- First term, \( a = 4 \)
- Last term, \( l = 31 \)
- Sum of terms, \( S_n = 175 \)
2. Use the sum formula to solve for the number of terms \( n \):
\[ 175 = \frac{n}{2}(4 + 31) \]
\[ 175 = \frac{n}{2}(35) \]
3. Multiply both sides by 2 to clear the fraction:
\[ 350 = 35n \]
\[ n = \frac{350}{35} = 10 \]
Thus, the number of terms in the AP is 10.
4. Now, use the last term formula to find the common difference \( d \):
\[ l = a + (n - 1)d \]
Substitute \( l = 31 \), \( a = 4 \), and \( n = 10 \):
\[ 31 = 4 + (10 - 1)d \]
\[ 31 = 4 + 9d \]
5. Isolate the term containing \( d \):
\[ 31 - 4 = 9d \]
\[ 27 = 9d \]
\[ d = \frac{27}{9} = 3 \]
Thus, the common difference is 3.

Step 4: Final Answer:
The number of terms \( n \) is 10 and the common difference \( d \) is 3.