The following sequence is given below:
1,3,3,3,5,5,5,5,5,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,……`
The sum of the 1st, 3rd , 8th, 15th , 24th , 35th, …. 99th` term will be :

Options

A. a. 100
B. b. 101
C. c. 121
D. d. 81

Answer

To solve the problem, we need to analyze the given sequence and find the sum of specific terms. The sequence is as follows:

1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, ...

### Step 1: Identify the Pattern in the Sequence
The sequence consists of odd numbers, where each odd number \( n \) appears \( n \) times. For example:
- The number 1 appears 1 time.
- The number 3 appears 3 times.
- The number 5 appears 5 times.
- The number 7 appears 7 times.
- The number 9 appears 9 times.
- This pattern continues with 11 appearing 11 times, 13 appearing 13 times, and so on.

### Step 2: Determine the Position of Each Term
To find the positions of the terms in the sequence, we can calculate how many terms there are up to each odd number:
- 1: 1 term (cumulative total = 1)
- 3: 3 terms (cumulative total = 1 + 3 = 4)
- 5: 5 terms (cumulative total = 4 + 5 = 9)
- 7: 7 terms (cumulative total = 9 + 7 = 16)
- 9: 9 terms (cumulative total = 16 + 9 = 25)
- 11: 11 terms (cumulative total = 25 + 11 = 36)
- 13: 13 terms (cumulative total = 36 + 13 = 49)
- 15: 15 terms (cumulative total = 49 + 15 = 64)
- 17: 17 terms (cumulative total = 64 + 17 = 81)
- 19: 19 terms (cumulative total = 81 + 19 = 100)

### Step 3: Identify the Required Terms
We need to find the 1st, 3rd, 8th, 15th, 24th, 35th, and 99th terms:
- **1st term**: 1
- **3rd term**: 3
- **8th term**: 5 (since the 8th term falls in the range of 5)
- **15th term**: 7 (since the 15th term falls in the range of 7)
- **24th term**: 9 (since the 24th term falls in the range of 9)
- **35th term**: 11 (since the 35th term falls in the range of 11)
- **99th term**: 19 (since the 99th term falls in the range of 19)

### Step 4: Calculate the Sum of the Identified Terms
Now, we sum these identified terms:
\[ 1 + 3 + 5 + 7 + 9 + 11 + 19 \]

Calculating the sum step by step:
- \( 1 + 3 = 4 \)
- \( 4 + 5 = 9 \)
- \( 9 + 7 = 16 \)
- \( 16 + 9 = 25 \)
- \( 25 + 11 = 36 \)
- \( 36 + 19 = 55 \)

Thus, the total sum is **55**.

### Final Answer:
The sum of the 1st, 3rd, 8th, 15th, 24th, 35th, and 99th terms is **55**.

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