How many terms are there in an AP whose first and fifth terms are -14 and 2 respectively and the sum of terms is 40

Options

A. 15
B. 10
C. 5
D. 20

Answer

To solve the problem step by step, we will follow the sequence of operations needed to find the number of terms in the arithmetic progression (AP).

### Step 1: Identify the known values
We know the following:
- First term (A) = -14
- Fifth term (A5) = 2
- Sum of terms (S) = 40

### Step 2: Use the formula for the nth term of an AP
The nth term of an AP can be expressed as:
\[ A_n = A + (n - 1) \cdot D \]
Where:
- \( A \) = first term
- \( D \) = common difference
- \( n \) = number of terms

For the fifth term:
\[ A_5 = A + (5 - 1) \cdot D \]
Substituting the known values:
\[ 2 = -14 + 4D \]

### Step 3: Solve for D
Rearranging the equation:
\[ 4D = 2 + 14 \]
\[ 4D = 16 \]
\[ D = \frac{16}{4} = 4 \]

### Step 4: Use the formula for the sum of the first n terms of an AP
The sum of the first n terms (S) is given by:
\[ S_n = \frac{n}{2} \cdot (2A + (n - 1)D) \]
Substituting the known values:
\[ 40 = \frac{n}{2} \cdot (2(-14) + (n - 1) \cdot 4) \]

### Step 5: Simplify the equation
\[ 40 = \frac{n}{2} \cdot (-28 + 4(n - 1)) \]
\[ 40 = \frac{n}{2} \cdot (-28 + 4n - 4) \]
\[ 40 = \frac{n}{2} \cdot (4n - 32) \]
Multiplying both sides by 2:
\[ 80 = n(4n - 32) \]
\[ 80 = 4n^2 - 32n \]

### Step 6: Rearrange into standard quadratic form
\[ 4n^2 - 32n - 80 = 0 \]
Dividing the entire equation by 4:
\[ n^2 - 8n - 20 = 0 \]

### Step 7: Solve the quadratic equation using the quadratic formula
The quadratic formula is:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -8, c = -20 \):
\[ n = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-20)}}{2 \cdot 1} \]
\[ n = \frac{8 \pm \sqrt{64 + 80}}{2} \]
\[ n = \frac{8 \pm \sqrt{144}}{2} \]
\[ n = \frac{8 \pm 12}{2} \]

### Step 8: Calculate the possible values for n
1. \( n = \frac{20}{2} = 10 \)
2. \( n = \frac{-4}{2} = -2 \) (not valid since n must be positive)

Thus, the number of terms in the AP is \( n = 10 \).

### Final Answer:
The number of terms in the AP is **10**.

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