The sum of all the terms of the A.P.7,10,13,... `l` is 1242. where `l` is the last term of the A.P. Find the value of `l`.

Options

A. 67
B. 79
C. 85
D. 102

Answer

To find the value of the last term \( l \) of the arithmetic progression (A.P.) given by \( 7, 10, 13, \ldots \) where the sum of all terms is \( 1242 \), we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is \( 7 \) and the common difference \( d \) can be calculated as:
\[ d = 10 - 7 = 3 \]

### Step 2: Use the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum of the first \( n \) terms \( S_n \) of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
We know \( S_n = 1242 \), so we can set up the equation:
\[ 1242 = \frac{n}{2} \times (2 \times 7 + (n-1) \times 3) \]

### Step 3: Simplify the equation
Substituting \( a = 7 \) and \( d = 3 \) into the equation:
\[ 1242 = \frac{n}{2} \times (14 + (n-1) \times 3) \]
This simplifies to:
\[ 1242 = \frac{n}{2} \times (14 + 3n - 3) \]
\[ 1242 = \frac{n}{2} \times (3n + 11) \]

### Step 4: Multiply both sides by 2 to eliminate the fraction
\[ 2484 = n(3n + 11) \]
This expands to:
\[ 3n^2 + 11n - 2484 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = 11 \), and \( c = -2484 \):
\[ n = \frac{-11 \pm \sqrt{11^2 - 4 \times 3 \times (-2484)}}{2 \times 3} \]
Calculating the discriminant:
\[ n = \frac{-11 \pm \sqrt{121 + 29712}}{6} \]
\[ n = \frac{-11 \pm \sqrt{29833}}{6} \]
Calculating \( \sqrt{29833} \approx 173 \):
\[ n = \frac{-11 \pm 173}{6} \]

### Step 6: Calculate the possible values for \( n \)
Calculating the two potential values for \( n \):
1. \( n = \frac{162}{6} = 27 \)
2. \( n = \frac{-184}{6} \) (not valid since \( n \) must be positive)

Thus, \( n = 27 \).

### Step 7: Find the last term \( l \)
Using the formula for the \( n \)-th term of an A.P.:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ l = 7 + (27-1) \times 3 \]
\[ l = 7 + 26 \times 3 \]
\[ l = 7 + 78 = 85 \]

### Final Answer
The value of \( l \) is \( 85 \).

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