WebWork Sequences 1) list of #'s 2) graph 3) closed form 4) recursive form
4) Write the first 8 terms of the following sequence: \(a_{1} = 3, a_{n+1}=a_{n}+2\)¶
\(\left\{3,5,7,9,11,13,15,17\right\}\)
5) Write the first 8 terms of the following sequence: \(a_{1}=1,\ a_{n+1}\ =\ 3a_{n}\)¶
\(\left\{1,3,9,27,81,243,729,2187\right\}\)
6) Give the following sequence in closed form: \(1,\ 8,\ 15,\ 22,\ 29,...\)¶
\(7n+1\)
Now give the sequence in recursive form.¶
\(a_{1}=1;a_{n+1}=a_{n}+7\)
7) Give the following sequence in closed form: \(1,\ 4,\ 9,\ 16,\ 25,\ 36,...\)¶
\(a_{n}=n^{2}\)
Now give the sequence in recursive form.¶
\(a_{1}=1;a_{n+1}=a_{n}+\left(2n+1\right)\)
8) Give the following sequence in closed form: \(1,\ 2,\ 6,\ 24,\ 120,\ 720,...\)¶
\(a_{n}=n!\)
Now give the sequence in recursive form.¶
\(a_{1}=1;a_{n+1}=a_{n}\left(n+1\right)\)
Summing Sequences \(\Sigma\)¶
\(a_n=2n+1\) (closed form)
\(a_n:3,5,7,9,11,\dots\)
\(n:1,2,3,4,5,\dots\)
\(\(\text{Format: }\sum_{n=\text{Start Index}}^{\text{end index}}{\text{(closed form)}}\)\)
\(\(\text{Example: }(5+7+9)=\sum_{n=2}^{4}{2n+1}\)\)
9) Write the following as a sum: \(\(\sum_{n=1}^{12}n\)\)¶
\(1+2+3+4+5+6+7+8+9+10+11+12\)
Now evaluate the sum.¶
\(78\)
10) Write the following as a sum: \(\(\sum_{n=2}^{5}n^{2}\)\)¶
\(4+9+16+25\)
Now evaluate the sum.¶
\(54\)
11) Write the following as a sum: \(\(\sum_{n=3}^{6}\left(n+1\right)\left(n+2\right)\)\)¶
\(20+30+42+56\)
Now evaluate the sum.¶
\(148\)
12) Write the following sum using summation notation: \(2+4+6+8+10+12\)¶
\[\sum_{n=1}^{6}2n\]
13) Write the following sum using summation notation: \(3+5+7+9\)¶
\[\sum_{n=1}^{4}2n+1\]
14) Write the following sum using summation notation: \(1+16+81\)¶
\[\sum_{n=1}^{3}n^{4}\]
15) Write the following sum using summation notation: \(1+5+9+13\)¶
\[\sum_{n=1}^{4}4n-3\]