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: Understanding the Question:
This question is from the topic "Arithmetic Progressions (A.P.)".
We are given the first term ($a$), the last term ($l$ or $a_n$), and the common difference ($d$) of a decreasing A.P.
We need to find the total number of terms ($n$) in this progression, and then compute the sum of all these $n$ terms ($S_n$).

Step 2: Key Formula or Approach:
1. Use the $n^{\text{th}}$ term formula to find the number of terms $n$:
\[ a_n = a + (n - 1)d \] 2. Once $n$ is calculated, compute the sum of these terms using the simplified arithmetic progression sum formula:
\[ S_n = \frac{n}{2} (a + l) \] where $l$ is the last term of the sequence.

Step 3: Detailed Explanation:

• Identify the given parameters:
- First term ($a$) $= 32$
- Last term ($l$ or $a_n$) $= -10$
- Common difference ($d$) $= -2$

• Apply the $n^{\text{th}}$ term formula to find $n$:
\[ a_n = a + (n - 1)d \] \[ -10 = 32 + (n - 1)(-2) \]

• Rearrange the equation to isolate the term with $n$:
\[ -10 - 32 = (n - 1)(-2) \] \[ -42 = -2(n - 1) \]

• Divide both sides of the equation by $-2$:
\[ n - 1 = \frac{-42}{-2} \] \[ n - 1 = 21 \] Add 1 to both sides:
\[ n = 22 \] So there are 22 terms in this progression.

• Now, use the sum formula for an A.P. to calculate $S_{22}$:
\[ S_n = \frac{n}{2} (a + l) \] \[ S_{22} = \frac{22}{2} (32 + (-10)) \]

• Calculate the value inside the parentheses and simplify:
\[ S_{22} = 11 \times (32 - 10) \] \[ S_{22} = 11 \times 22 \] \[ S_{22} = 242 \]

Step 4: Final Answer:
The number of terms in the A.P. is $22$, and their sum is $242$.