A sequence is shown below :
1,4,-2,1…..
The first term is 1. The third term is obtained by dividing the second term by (-2) and the fourth term is obtained by adding 3 to the third term. The same above cycle then repeats for the `5^(th), 6^(th)` and `7^(th)` terms and so on. What is the sum of the first 22 term of the above sequence ?

Options

A. 0.08
B. 21
C. 22
D. 0.32

Answer

To solve the problem step by step, we will first identify the repeating pattern in the sequence and then calculate the sum of the first 22 terms.

### Step 1: Identify the sequence pattern
The sequence provided is:
1. First term (T1) = 1
2. Second term (T2) = 4
3. Third term (T3) = T2 / (-2) = 4 / (-2) = -2
4. Fourth term (T4) = T3 + 3 = -2 + 3 = 1

So, the first four terms are: 1, 4, -2, 1.

### Step 2: Recognize the repeating cycle
From the calculations, we can see that the sequence repeats every 3 terms after the first four:
- T1 = 1
- T2 = 4
- T3 = -2
- T4 = 1
- T5 = 4
- T6 = -2
- T7 = 1
- T8 = 4
- T9 = -2
- ...

The repeating cycle is: 1, 4, -2.

### Step 3: Count the occurrences of each term in the first 22 terms
Now, we need to determine how many times each term appears in the first 22 terms.

- The sequence has 4 unique terms in the first cycle (1, 4, -2, 1) and then repeats every 3 terms.
- For the first 22 terms:
- The first term is 1 (occurs once).
- The next 3 terms (T2, T3, T4) are 4, -2, and 1.
- This pattern (1, 4, -2) continues for the next terms.

To find how many times each term appears:
- The sequence has 22 terms, and the first term is counted separately.
- The repeating part (4, -2, 1) occurs 18 times (22 - 4 = 18).
- The number of complete cycles of (4, -2, 1) in 18 terms is 6 cycles (18 / 3 = 6).
- Therefore, in 22 terms:
- 1 appears: 1 (from T1) + 6 (from T4, T7, T10, T13, T16, T19) = 7 times.
- 4 appears: 6 (from T2, T5, T8, T11, T14, T17) = 7 times.
- -2 appears: 6 (from T3, T6, T9, T12, T15, T18) = 6 times.

### Step 4: Calculate the sum of the first 22 terms
Now we can calculate the total sum:
- Total sum = (Number of 1s × 1) + (Number of 4s × 4) + (Number of -2s × -2)
- Total sum = (7 × 1) + (7 × 4) + (6 × -2)
- Total sum = 7 + 28 - 12
- Total sum = 23.

### Final Answer
The sum of the first 22 terms of the sequence is **23**.

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