3. Odd numbers take the form [tex]2k-1[/tex] for [tex]k=1,2,3,\ldots[/tex]. So the sum of the first [tex]n[/tex] (positive) odd integers is
[tex]\displaystyle\sum_{k=1}^n(2k-1)[/tex]
Recall that
[tex]\displaystyle\sum_{k=1}^n1=n[/tex]
[tex]\displaystyle\sum_{k=1}^nk=\frac{n(n+1)}2[/tex]
[tex]\implies\displaystyle\sum_{k=1}^n(2k-1)=n(n+1)-n=n^2[/tex]
4. The sequence is geometric with common ratio [tex]r=-\frac32[/tex]:
[tex]\dfrac89\left(-\dfrac32\right)=-\dfrac43[/tex]
[tex]-\dfrac43\left(-\dfrac32\right)=2[/tex]
and so on. Then the [tex]n[/tex]th term in the sequence is given by the rule
[tex]a_n=\left(-\dfrac32\right)^{n-1}\dfrac89[/tex]
and so the 14th term in the sequence is
[tex]a_{14}=\left(-\dfrac32\right)^{13}\dfrac89=-\dfrac{177,147}{1024}[/tex]
5. The sequence is geometric with ratio [tex]r=\frac12[/tex], so that the [tex]n[/tex]th term in the sequence is
[tex]a_n=\left(\dfrac12\right)^{n-1}8[/tex]
The sum of the first 10 terms is
[tex]S_{10}=\displaystyle\sum_{k=1}^{10}a_k=8\sum_{k=1}^{10}\frac1{2^{k-1}}[/tex]
We then have
[tex]S_{10}=8+4+2+\cdots+\dfrac1{64}[/tex]
[tex]2S_{10}=16+8+4+\cdots+\dfrac1{32}[/tex]
[tex]\implies S_{10}-2S_{10}=-S_{10}=\dfrac1{64}-16\implies S_{10}=16-\dfrac1{64}=\dfrac{1023}{64}[/tex]
6. The [tex]n[/tex]th term of the sequence {1/2, 1/4, 1/12, ...} is given by
[tex]a_n=\dfrac1{2n!}[/tex]
so the given sum can be written as
[tex]2+\dfrac12\displaystyle\sum_{n=1}^\infty\frac1{n!}[/tex]
Recall that
[tex]e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}[/tex]
so the given sum is equal to
[tex]2+\dfrac{e-1}2=\dfrac{3+e}2[/tex]
