Question

Solve any one of the following internal choices (a) or (b):
22(a) In an A.P., the first term is 32 and the last term is -10. If the common difference is -2, then find the number of terms and their sum.

Hint:

When both the first term and the last term are known, always use the formula $S_n = \frac{n}{2}(a+l)$ instead of $S_n = \frac{n}{2}[2a + (n-1)d]$.
This saves simple arithmetic calculation steps and reduces the chance of making computational mistakes!

Answer 1

Step 1: Find the number of terms by counting how many steps of -2 it takes to reach the last term, instead of plugging into the nth-term formula right away.
We start at $32$ and take steps of $-2$ each time until we reach $-10$. The total distance covered is:
\[ 32 - (-10) = 42 \]
Since each step covers a distance of $2$ (the size of the common difference, ignoring sign), the number of steps taken is:
\[ \frac{42}{2} = 21 \]
Since 21 steps are needed to go from the first term to the last term, the total number of terms is one more than the number of steps:
\[ n = 21 + 1 = 22 \]

Step 2: Use the other standard sum formula, based on the first term and common difference, instead of the first-term-and-last-term version.
\[ S_n = \frac{n}{2}\left[2a + (n-1)d\right] \]

Step 3: Substitute the known values.
\[ S_{22} = \frac{22}{2}\left[2(32) + (22-1)(-2)\right] \]

Step 4: Simplify inside the brackets first.
\[ 2(32) = 64 \]
\[ (21)(-2) = -42 \]
\[ 64 + (-42) = 22 \]

Step 5: Multiply out to get the sum.
\[ S_{22} = 11 \times 22 = 242 \]

Final Answer:
The number of terms is $22$, and their sum is $242$. \[ \boxed{n = 22,\ S_n = 242} \]