The first and the last terms of an A.P. are 34 and 700 respectively. If the common difference is 18, how many terms are there and what is their sum?

Answer

To solve the problem step by step, we will determine the number of terms in the arithmetic progression (A.P.) and then calculate their sum.

### Step 1: Identify the given values
- First term (a) = 34
- Last term (l) = 700
- Common difference (d) = 18

### 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:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Set up the equation for the last term
Since the last term \( a_n \) is given as 700, we can set up the equation:
\[ 700 = 34 + (n - 1) \cdot 18 \]

### Step 4: Solve for \( n \)
Rearranging the equation gives:
\[ 700 - 34 = (n - 1) \cdot 18 \]
\[ 666 = (n - 1) \cdot 18 \]

Now, divide both sides by 18:
\[ n - 1 = \frac{666}{18} \]
\[ n - 1 = 37 \]

Adding 1 to both sides gives:
\[ n = 37 + 1 = 38 \]

So, the number of terms \( n \) is 38.

### Step 5: Calculate the sum of the first n terms
The formula for the sum \( S_n \) of the first n terms of an A.P. is:
\[ S_n = \frac{n}{2} \cdot (a + l) \]

Substituting the values we have:
\[ S_{38} = \frac{38}{2} \cdot (34 + 700) \]
\[ S_{38} = 19 \cdot 734 \]

Now, calculate:
\[ S_{38} = 19 \cdot 734 = 13946 \]

### Final Answers
- The total number of terms is **38**.
- The sum of the first 38 terms is **13946**.

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