Practice section 4 of 17

(a) Why is showing only $u_2/u_1=u_3/u_2$ insufficient to conclude that a sequence is a GP? (b) Is the constant sequence $2,2,2,\ldots,2$ an AP or a GP?

For each series, find an expression for its nth term $u_n$ and determine whether the terms form an AP or GP: (a) $S_n=2^n-1$; (b) $S_n=n^2+3n$; (c) $S_n=n^2+3n+1$, where $S_n$ is the sum of the first $n$ terms.

Write out each sum in full and state its number of terms: (i) $\sum_{r=1}^{6}r$; (ii) $\sum_{r=5}^{8}r^2$; (iii) $\sum_{k=0}^{4}(2k-1)$; (iv) $\sum_{r=0}^{\infty}(1/2)^r$.

Find an expression for the nth term and express each series in sigma notation: (a) $1+4+7+10+\cdots+88$; (b) $1+3+5+7+\cdots+99$; (c) $-2+3-4+5-\cdots+41$.

Express in sigma notation: (i) $1^4+2^4+\cdots+n^4+(n+1)^4$; (ii) $-1+2-3+4-5+6$.

Evaluate $\sum_{r=2}^{8}r^4(r-1)$ using the calculator summation command illustrated in the source.