The ratio of the fifth and the sixth terms of the sequence 1 ,3 , 6 , 10, …………is

Options

A. `5:6`
B. `5:7`
C. `7:5`
D. `6:5`

Answer

To find the ratio of the fifth and sixth terms of the sequence 1, 3, 6, 10, ..., we first need to identify the pattern in the sequence.

### Step-by-Step Solution:

1. **Identify the Sequence**: The given sequence is 1, 3, 6, 10.
- The first term (T1) is 1.
- The second term (T2) is 3.
- The third term (T3) is 6.
- The fourth term (T4) is 10.

2. **Find the Differences**: Let's find the differences between consecutive terms:
- T2 - T1 = 3 - 1 = 2
- T3 - T2 = 6 - 3 = 3
- T4 - T3 = 10 - 6 = 4

The differences are 2, 3, and 4. We can see that the difference between terms is increasing by 1 each time.

3. **Calculate the Fifth Term (T5)**:
- The difference to find the fifth term will be 5 (the next number in the difference sequence).
- So, T5 = T4 + 5 = 10 + 5 = 15.

4. **Calculate the Sixth Term (T6)**:
- The difference to find the sixth term will be 6 (the next number in the difference sequence).
- So, T6 = T5 + 6 = 15 + 6 = 21.

5. **Find the Ratio of the Fifth and Sixth Terms**:
- Now we have T5 = 15 and T6 = 21.
- The ratio of the fifth term to the sixth term is T5 : T6 = 15 : 21.

6. **Simplify the Ratio**:
- To simplify 15 : 21, we can divide both numbers by their greatest common divisor, which is 3.
- So, 15 ÷ 3 = 5 and 21 ÷ 3 = 7.
- Therefore, the simplified ratio is 5 : 7.

### Final Answer:
The ratio of the fifth and sixth terms of the sequence is 5 : 7.

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