Question

The first term and the last term of an A.P. are 8 and 341 respectively. If the common difference is 9, then find the number of terms in it and their sum.

Hint:

When the last term is known, the $S_n = \frac{n}{2}(a+l)$ formula is much faster than the $S_n = \frac{n}{2}[2a+(n-1)d]$ formula.

Answer 1

Step 1: Write the given information.
The given arithmetic progression (A.P.) has:
First term \(a = 8\)
Last term \(l = 341\)
Common difference \(d = 9\)

We need to find:
1) The number of terms (n)
2) The sum of all terms.

Step 2: Use the formula for the last term of an A.P.
The formula for the last term is:
\(l = a + (n-1)d\)

Substitute the given values:
\(341 = 8 + (n-1) \times 9\)

Step 3: Simplify the equation.
\(341 - 8 = (n-1) \times 9\)
\(333 = 9(n-1)\)
\(\frac{333}{9} = n-1\)
\(37 = n-1\)
\(n = 38\)

Thus, the number of terms is 38.

Step 4: Use the formula to find the sum of n terms.
The formula for the sum of n terms of an A.P. is:
\(S_n = \frac{n}{2}(a + l)\)

Substitute the values:
\(S_{38} = \frac{38}{2}(8 + 341)\)

Step 5: Simplify the expression.
\(S_{38} = 19 \times 349\)
\(S_{38} = 6631\)

Final Answer:
Number of terms \(n = 38\)
Sum of all terms \(S_{38} = 6631\).