91Ó°ÊÓ

Evaluate \(\lim _{n \rightarrow \infty} n \ln \left(1+\frac{1}{n}\right)\).

A formula is given for the \(n^{\text {th }}\) term of a sequence \(a_{1}, a_{2}, \ldots\) (a) Write the sequence using the three-dot notation, giving the first four terms. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ a_{n}=1-\frac{1}{3 n} $$

Evaluate the arithmetic series. $$ \sum_{k=5}^{65}(4 k-1) $$

Consider an arithmetic sequence with first term \(b\) and difference \(d\) between consecutive terms (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=2, d=5 $$

Evaluate \(\lim _{n \rightarrow \infty} n\left(\ln \left(3+\frac{1}{n}\right)-\ln 3\right)\).

Evaluate \(\lim _{n \rightarrow \infty} n\left(\ln \left(3+\frac{1}{n}\right)-\ln 3\right)\)

Consider an arithmetic sequence with first term \(b\) and difference \(d\) between consecutive terms (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=7, d=3 $$

Evaluate the arithmetic series. $$ \sum_{k=10}^{900}(3 k-2) $$

Find the sum of all the four-digit positive integers.

Consider an arithmetic sequence with first term \(b\) and difference \(d\) between consecutive terms (a) Write the sequence using the three-dot notation, giving the first four terms of the sequence. (b) Give the \(100^{\text {th }}\) term of the sequence. $$ b=4, d=-6 $$