Question:medium

If \(S_n\) denotes the sum of the first \(n\) terms of an Arithmetic Progression, and \(S_1 : S_4 = 1 : 10\), then the ratio of the first term to the fourth term is:

Show Hint

Write S1 = a and S4 = 4a + 6d using the AP sum formula, then use the given ratio S1:S4 = 1:10 to relate a and d.
Updated On: Jul 13, 2026
  • 1 : 3
  • 2 : 3
  • 1 : 4
  • 1 : 5
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Pick convenient numbers that satisfy the given ratio, instead of working with letters throughout.
We are told $S_1 : S_4 = 1:10$. Since $S_1$ is always just the first term, assume the first term is $a = 1$ so that $S_1 = 1$, and work out what $d$ must be so that $S_4 = 10$.

Step 2: Write $S_4$ using the sum formula.
\[ S_4 = \frac{4}{2}(2a + 3d) = 2(2(1) + 3d) = 2(2+3d) = 4 + 6d \]

Step 3: Set this equal to 10 and solve for $d$.
\[ 4 + 6d = 10 \]
\[ 6d = 6 \]
\[ d = 1 \]
So with $a=1$, the common difference is also $1$, giving the AP $1, 2, 3, 4, 5, \ldots$

Step 4: Check this AP against the original ratio.
$S_1 = 1$. $S_4 = 1+2+3+4 = 10$. This gives $S_1:S_4 = 1:10$, confirming this AP fits the given condition.

Step 5: Read off the first and fourth terms directly from this AP.
First term $= 1$. Fourth term $= 4$.
\[ \frac{T_1}{T_4} = \frac{1}{4} \]

Step 6: Confirm the answer holds in general, not just for this one example.
The relation $a=d$ found this way holds for any starting value of $a$, since scaling $a$ scales $d$ in the same proportion. So the ratio $1:4$ is the answer for every AP satisfying the given condition, not only the one picked here.

Final Answer:
\[ \boxed{1:4} \]
Was this answer helpful?
0

Top Questions on Arithmetic Progression

Questions Asked in XAT exam

Read the scenario and answer the question:
The upper hinge of a dataset is the median of all the values to the right of the median of the dataset in an ascending arrangement, while the lower hinge of the dataset is the median of all the values to the left of the median of the dataset in the same arrangement.
For example, consider the dataset 4, 3, 2, 6, 4, 2, 7. When arranged in the ascending order, it becomes 2, 2, 3, 4, 4, 6, 7. The median is 4 (the bold value), and hence the upper hinge is the median of 4, 6, 7, i.e., 6. Similarly, the lower hinge is 2. 
A student has surveyed thirteen of her teachers, and recorded their work experience (in integer years). Two of the values recorded by the student got smudged, and she cannot recall those values. All she remembers is that those two values were unequal, so let us write them as A and B, where A < B. The remaining eleven values, as recorded, are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29. Moreover, the student also remembers the following summary measures, calculated based on all the thirteen values: 
Minimum: 2
Lower Hinge: 6.5 
Median: 12 
Upper Hinge: 21 
Maximum: 29
View Solution
Read the following scenario and answer the questions that follow.
A T20 cricket match consists of two teams playing twenty overs each, numbered 1 to 20. The runs scored in any over is a non-negative integer. The run rate at the end of any over is the average runs scored up to and including that over, i.e., the run rate at the end of the kth over is the average number of runs scored in overs numbered \(1, 2, …, k,\) where \(1 \leq k \leq 20, k\) a positive integer.
The following table indicates the run rate of a team at the end of some of the overs during a T20 cricket match (correct up to 2 decimal places), where \(1 \leq N – 2 < N + 6 \leq 20\), N a positive integer. It is also known that the team did not score less than 6 runs and more than 15 runs in any over.
Over NumberRun Rate
N-28.00
N7.43
N+28.11
N+48.45
N+68.08
View Solution