The common difference, last term and sum of terms of an A.P. are 4, 31, and 136 respectively. Find the number of terms.

Explanation

To solve the problem step by step, we need to find the number of terms in the arithmetic progression (A.P.) given the common difference (d), last term (l), and the sum of the terms (S).

### Step 1: Write down the given information
- Common difference, \( d = 4 \)
- Last term, \( l = 31 \)
- Sum of terms, \( S = 136 \)

### Step 2: Use the formula for the last term of an A.P.
The last term of an A.P. can be expressed as:
\[ l = A + (N - 1) \cdot d \]
where \( A \) is the first term and \( N \) is the number of terms. Substituting the known values:
\[ 31 = A + (N - 1) \cdot 4 \]
Rearranging gives:
\[ A + 4N - 4 = 31 \]
Thus:
\[ A + 4N = 35 \quad \text{(Equation 1)} \]

### Step 3: Use the formula for the sum of the first N terms of an A.P.
The sum of the first N terms of an A.P. is given by:
\[ S = \frac{N}{2} \cdot (2A + (N - 1) \cdot d) \]
Substituting the known values:
\[ 136 = \frac{N}{2} \cdot (2A + (N - 1) \cdot 4) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 272 = N \cdot (2A + 4N - 4) \]
This simplifies to:
\[ 272 = N \cdot (2A + 4N - 4) \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we have \( A = 35 - 4N \). Substitute this into Equation 2:
\[ 272 = N \cdot (2(35 - 4N) + 4N - 4) \]
Expanding this:
\[ 272 = N \cdot (70 - 8N + 4N - 4) \]
This simplifies to:
\[ 272 = N \cdot (66 - 4N) \]
Rearranging gives:
\[ 272 = 66N - 4N^2 \]
Rearranging further:
\[ 4N^2 - 66N + 272 = 0 \]

### Step 5: Simplify the quadratic equation
Dividing the entire equation by 2:
\[ 2N^2 - 33N + 136 = 0 \]

### Step 6: Solve the quadratic equation
To solve \( 2N^2 - 33N + 136 = 0 \), we can use the middle term splitting method:
\[ 2N^2 - 16N - 17N + 136 = 0 \]
Factoring gives:
\[ 2N(N - 8) - 17(N - 8) = 0 \]
Factoring out \( (N - 8) \):
\[ (2N - 17)(N - 8) = 0 \]

### Step 7: Find the values of N
Setting each factor to zero:
1. \( 2N - 17 = 0 \) gives \( N = \frac{17}{2} = 8.5 \) (not valid since N must be an integer)
2. \( N - 8 = 0 \) gives \( N = 8 \) (valid)

### Conclusion
The number of terms in the A.P. is \( N = 8 \).