Question

In an A.P., the first term is 3 and the last term is 17. The sum of all the terms in the sequence is 70. Then the number of terms in the arithmetic sequence is

• A. 7
• B. 5
• C. 9
• D. 6
• E. 8

Hint:

Formula Tip: Whenever a sequence problem provides you with the first term, the last term, and the sum, immediately jump to $S_n = \frac{n}{2}(a + l)$. Do not waste time trying to calculate the common difference $d$.

Answer 1

The Correct Option is A

Concept:
The sum of a finite Arithmetic Progression (A.P.) can be calculated using the first and last terms without needing to know the common difference. The formula is $S_n = \frac{n}{2}(a + l)$, where $S_n$ is the total sum, $n$ is the number of terms, $a$ is the first term, and $l$ is the last term.

Step 1: Identify the given variables.
From the problem description, extract the known values: First term: $a = 3$ Last term: $l = 17$ Sum of terms: $S_n = 70$

Step 2: State the appropriate formula.
Since we know the first and last terms, we select the sum formula: $$S_n = \frac{n}{2}(a + l)$$

Step 3: Substitute the known values into the equation.
Plug the values from Step 1 into the selected formula: $$70 = \frac{n}{2}(3 + 17)$$

Step 4: Simplify the equation.
Add the terms inside the parentheses and multiply: $$70 = \frac{n}{2}(20)$$ $$70 = 10n$$

Step 5: Solve for the number of terms (n).
Isolate $n$ by dividing both sides by 10: $$n = \frac{70}{10}$$ $$n = 7$$ Hence the correct answer is (A) 7.