Final answer:
The given series S = 3n + 4n² represents the sum of the first n terms of an arithmetic progression, as proven by the difference between consecutive terms being constant, specifically 7.
Step-by-step explanation:
To show that the series S = 3n + 4n² represents the sum of the first n terms of an arithmetic progression (AP), we must demonstrate that the difference between consecutive terms is constant. To find the nth term of this series (an), we calculate the sum up to n terms and subtract the sum up to (n-1) terms.
Let's first consider the sum to (n-1) terms, which is Sn-1 = 3(n-1) + 4(n-1)². To find an, we subtract Sn-1 from Sn:
an = Sn - Sn-1
= (3n + 4n²) - [3(n-1) + 4(n-1)²]
= 3n + 4n² - (3n - 3 + 4n² - 8n + 4)
= 3n + 4n² - 3n + 3 - 4n² + 8n - 4
= 7n - 1
This is an arithmetic sequence because the difference between consecutive terms is constant and equal to 7. Hence, it's confirmed that the given series represents an arithmetic progression.