If the sum of first n terms of an A.P. is given by \( S_n = \frac{n}{2}(3n+1) \), then the first term of the A.P. is

Question

In which of the following situations, does the list of numbers involved make as arithmetic progression and why?
(1) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.
(2) The amount of air present in a cylinder when a vacuum pump removes \(\frac{1}{4}\) of the air remaining in the cylinder at a time.
(3) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.
(4) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8% per annum.

Answer 1

Given:
The sum of the first \(n\) terms of an arithmetic progression (A.P.) is:
\[ S_n = \frac{n}{2}(3n + 1) \] where \(S_n\) denotes the sum of the first \(n\) terms.

Step 1: Determine the objective
Find the first term \(a\) of the A.P.

Step 2: Relation between sum and first term
The first term \(a\) equals the sum of the first term, i.e.,
\[ a = S_1 \] Therefore, substituting \(n = 1\) in \(S_n\) yields the first term \(a\).

Step 3: Substitute \(n = 1\)
\[ S_1 = \frac{1}{2} (3 \times 1 + 1) = \frac{1}{2} (3 + 1) = \frac{1}{2} \times 4 = 2 \]

Step 4: Result Interpretation
\(S_1 = 2\) implies \(a = 2\).

Final Answer:
\[ \boxed{2} \]

If the sum of first n terms of an A.P. is given by \( S_n = \frac{n}{2}(3n+1) \), then the first term of the A.P. is

The Correct Option is A

Solution and Explanation

Top Questions on Arithmetic Progression

Questions Asked in CBSE Class X exam