The sum of how many terms of the A.P. 10,12,14,… will be 190 ?

Answer

To find the number of terms \( n \) in the arithmetic progression (A.P.) 10, 12, 14, ... such that their sum equals 190, we can use the formula for the sum of the first \( n \) terms of an A.P.:

\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 1: Identify the first term and common difference
From the given A.P.:
- First term \( a = 10 \)
- Common difference \( d = 12 - 10 = 2 \)

### Step 2: Set up the equation for the sum
We know that the sum \( S_n = 190 \). Substituting the values of \( a \) and \( d \) into the sum formula gives:

\[ 190 = \frac{n}{2} \times (2 \times 10 + (n - 1) \times 2) \]

### Step 3: Simplify the equation
Now, simplify the equation:

\[ 190 = \frac{n}{2} \times (20 + 2(n - 1)) \]

This simplifies to:

\[ 190 = \frac{n}{2} \times (20 + 2n - 2) \]

\[ 190 = \frac{n}{2} \times (2n + 18) \]

### Step 4: Eliminate the fraction
Multiply both sides by 2 to eliminate the fraction:

\[ 380 = n(2n + 18) \]

### Step 5: Rearrange the equation
Rearranging gives:

\[ 2n^2 + 18n - 380 = 0 \]

### Step 6: Simplify the quadratic equation
Dividing the entire equation by 2:

\[ n^2 + 9n - 190 = 0 \]

### Step 7: Factor the quadratic equation
Now we need to factor this quadratic equation. We look for two numbers that multiply to \(-190\) and add to \(9\). The numbers \(19\) and \(-10\) work:

\[ (n + 19)(n - 10) = 0 \]

### Step 8: Solve for \( n \)
Setting each factor to zero gives:

\[ n + 19 = 0 \quad \text{or} \quad n - 10 = 0 \]

This results in:

\[ n = -19 \quad \text{or} \quad n = 10 \]

Since \( n \) must be a positive integer, we have:

\[ n = 10 \]

### Conclusion
The sum of the first 10 terms of the A.P. is 190.

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