What could be a correct representation of the nth term for the sequence if the first term is 1 and each term thereafter involves increasing by a growing amount?

What could be a correct representation of the nth term for the sequence if the first term is 1 and each term thereafter involves increasing by a growing amount?

• A. n
• B. n^2
• C. 1 + (n(n-1))/2
• D. n+1

Answer: (C)

The nth term in this sequence reflects a quadratic pattern, where each term increases by an additional amount compared to the previous one. The first term is 1, and as the terms progress, they increase by increasingly larger amounts: the second term increases by 1, the third term increases by 2, the fourth by 3, and so on. This suggests a pattern similar to how triangular numbers are calculated. The formula provided in the correct choice, 1 + (n(n-1))/2, accurately accounts for this growing increment because it can be broken down as follows: 1. The term (n(n-1))/2 represents the sum of the first (n-1) integers, which itself is the growing total that corresponds to these increasing amounts. 2. When n=1, the formula gives us 1, which is our first term. 3. For n=2, it results in 1 + (2(1))/2 = 1 + 1 = 2, which is the second term. 4. For n=3, it results in 1 + (3(2))/2 = 1 + 3 = 4, which is the third term. 5. Continuing this

The nth term in this sequence reflects a quadratic pattern, where each term increases by an additional amount compared to the previous one. The first term is 1, and as the terms progress, they increase by increasingly larger amounts: the second term increases by 1, the third term increases by 2, the fourth by 3, and so on. This suggests a pattern similar to how triangular numbers are calculated.

The formula provided in the correct choice, 1 + (n(n-1))/2, accurately accounts for this growing increment because it can be broken down as follows:

1. The term (n(n-1))/2 represents the sum of the first (n-1) integers, which itself is the growing total that corresponds to these increasing amounts.

2. When n=1, the formula gives us 1, which is our first term.

3. For n=2, it results in 1 + (2(1))/2 = 1 + 1 = 2, which is the second term.

4. For n=3, it results in 1 + (3(2))/2 = 1 + 3 = 4, which is the third term.

5. Continuing this