Question

In parts (a)-(d), find the values of the sums in terms of n. In part (e), evaluate the limit of the sum from part (d).

a. n^2 + n
b. n(n + 1)
c. 2n
d. (n^2 + n)/2

Answer 1

In part (a), the sum is equal to n².

Step-by-step explanation:

In part (a), the sum is equal to n². To see this, imagine taking (n - 1) from the last term and adding it to the first term:

2[1 + (n - 1) + 3 + ... + (2n - 3) + (2n - 1) - (n - 1)]

This simplifies to 2[n + 3 + ... + (2n - 3) + n]. Now, take (n - 3) from the penultimate term and add it to the second term:

2[n + n +...+ n + n]

This gives us the final expression: 2n².