The 2nd, 31st and last term of an A.P. are `7(3)/(4) , (1)/(2) ` and `-6(1)/(2)` respectively. Find the first term and the number of terms.

Answer

To solve the problem, we need to find the first term and the number of terms in the Arithmetic Progression (A.P.) given the second term, the 31st term, and the last term. Let's denote the first term as \( a \) and the common difference as \( d \).

### Step 1: Write the equations for the given terms
The second term of an A.P. can be expressed as:
\[ a + d = 7 \frac{3}{4} = \frac{31}{4} \quad \text{(Equation 1)} \]

The 31st term can be expressed as:
\[ a + 30d = \frac{1}{2} \quad \text{(Equation 2)} \]

The last term is given as:
\[ a + (n-1)d = -6 \frac{1}{2} = -\frac{13}{2} \quad \text{(Equation 3)} \]

### Step 2: Solve Equations 1 and 2
From Equation 1:
\[ a + d = \frac{31}{4} \]

From Equation 2:
\[ a + 30d = \frac{1}{2} \]

Now, we can subtract Equation 1 from Equation 2:
\[ (a + 30d) - (a + d) = \frac{1}{2} - \frac{31}{4} \]

This simplifies to:
\[ 29d = \frac{1}{2} - \frac{31}{4} \]

Finding a common denominator (4):
\[ \frac{1}{2} = \frac{2}{4} \]
So,
\[ 29d = \frac{2}{4} - \frac{31}{4} = \frac{2 - 31}{4} = \frac{-29}{4} \]

Now, divide both sides by 29:
\[ d = \frac{-29}{4 \cdot 29} = -\frac{1}{4} \]

### Step 3: Substitute \( d \) back into Equation 1 to find \( a \)
Now substitute \( d \) back into Equation 1:
\[ a + d = \frac{31}{4} \]
\[ a - \frac{1}{4} = \frac{31}{4} \]

Adding \( \frac{1}{4} \) to both sides:
\[ a = \frac{31}{4} + \frac{1}{4} = \frac{32}{4} = 8 \]

### Step 4: Find the number of terms \( n \)
Now we can use Equation 3 to find \( n \):
\[ a + (n-1)d = -\frac{13}{2} \]
Substituting \( a = 8 \) and \( d = -\frac{1}{4} \):
\[ 8 + (n-1)(-\frac{1}{4}) = -\frac{13}{2} \]

This simplifies to:
\[ 8 - \frac{n-1}{4} = -\frac{13}{2} \]

Now, convert 8 to a fraction with a denominator of 4:
\[ 8 = \frac{32}{4} \]
So,
\[ \frac{32}{4} - \frac{n-1}{4} = -\frac{13}{2} \]

Finding a common denominator on the right side (which is 4):
\[ -\frac{13}{2} = -\frac{26}{4} \]
Thus, we have:
\[ \frac{32 - (n-1)}{4} = -\frac{26}{4} \]

Multiplying through by 4:
\[ 32 - (n-1) = -26 \]

Rearranging gives:
\[ 32 + 26 = n - 1 \]
\[ n - 1 = 58 \quad \Rightarrow \quad n = 59 \]

### Final Answer
The first term \( a \) is \( 8 \) and the number of terms \( n \) is \( 59 \).