The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum ?

Answer

To solve the problem step by step, we will follow the process of finding the number of terms in the arithmetic progression (A.P.) and then calculating the sum of those terms.

### Step 1: Identify the given values
- First term (A) = 17
- Last term (L) = 350
- Common difference (D) = 9

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ L = A + (N - 1) \times D \]
Where:
- L = last term
- A = first term
- N = number of terms
- D = common difference

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 350 = 17 + (N - 1) \times 9 \]

### Step 4: Simplify the equation
First, subtract 17 from both sides:
\[ 350 - 17 = (N - 1) \times 9 \]
\[ 333 = (N - 1) \times 9 \]

### Step 5: Solve for (N - 1)
Now, divide both sides by 9:
\[ N - 1 = \frac{333}{9} \]
\[ N - 1 = 37 \]

### Step 6: Solve for N
Now, add 1 to both sides to find N:
\[ N = 37 + 1 \]
\[ N = 38 \]

### Step 7: Calculate the sum of the A.P.
To find the sum (S) of the first N terms of an A.P., we can use the formula:
\[ S_N = \frac{N}{2} \times (A + L) \]
Where:
- S_N = sum of the first N terms
- A = first term
- L = last term
- N = number of terms

### Step 8: Substitute the values into the sum formula
Substituting the known values:
\[ S_{38} = \frac{38}{2} \times (17 + 350) \]

### Step 9: Simplify the sum calculation
Calculate:
\[ S_{38} = 19 \times (367) \]
\[ S_{38} = 6973 \]

### Final Answer
- The number of terms (N) is **38**.
- The sum of the terms (S) is **6973**.

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