# Q. What is the sum of the first 28 terms in the following sequence?

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, ..

a) 83

b) 84

c) 85

d) 86

Correct answer: b) 84

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

## To find the sum of the first 28 terms of the sequence:

1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, …

**Step 1: Understand the Pattern**

First, let’s observe the pattern in the sequence:

The sequence is built in groups, where each group starts with 1 and counts up to a certain number, then counts down to 2.

Here’s how the groups look:

Group 1: 1

Group 2: 1, 2

Group 3: 1, 3, 2

Group 4: 1, 4, 3, 2

Group 5: 1, 5, 4, 3, 2

Group 6: 1, 6, 5, 4, 3, 2

Group 7: 1, 7, 6, 5, 4, 3, 2

Each group n has n terms.

### Step 2: Sum the Terms in Each Group

Now, let’s sum the terms in each group:

Sum = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

**Step 3: Calculate the Total Sum**

Add up the sums from each group:

Total Sum = 1 + 3 + 6 + 10 + 15 + 21 + 28 = 84