a) Insert 6 arithmetic means between 15 and 27. b) Determine the common ratio of a geometric series whose first term and the fifth terms are 2 and 512 respectively.

a) Insert 6 arithmetic means between 15 and 27

Solution:

To insert 6 arithmetic means between 15 and 27, we need to construct an arithmetic sequence with 8 terms (since 6 means + 2 original terms = 8 terms).

The general form of an arithmetic sequence is: $$a,a+d,a+2d,\dots ,a+(n-1)d$$ where:

• $a$ = first term = 15
• $l$ = last term = 27
• $n$ = total number of terms = 8
• $d$ = common difference

The formula for the $n$-th term of an arithmetic sequence is: $$l=a+(n-1)d$$

Substituting the known values: $$27=15+(8-1)d$$ $$27=15+7d$$ $$27-15=7d$$ $$12=7d$$ $$d=\frac{12}{7}$$

Now, the 8 terms of the arithmetic sequence are:

1. $a_1=15$
2. $a_2=15+d=15+\frac{12}{7}=\frac{105}{7}+\frac{12}{7}=\frac{117}{7}$
3. $a_3=15+2d=15+2\times \frac{12}{7}=\frac{105}{7}+\frac{24}{7}=\frac{129}{7}$
4. $a_4=15+3d=15+3\times \frac{12}{7}=\frac{105}{7}+\frac{36}{7}=\frac{141}{7}$
5. $a_5=15+4d=15+4\times \frac{12}{7}=\frac{105}{7}+\frac{48}{7}=\frac{153}{7}$
6. $a_6=15+5d=15+5\times \frac{12}{7}=\frac{105}{7}+\frac{60}{7}=\frac{165}{7}$
7. $a_7=15+6d=15+6\times \frac{12}{7}=\frac{105}{7}+\frac{72}{7}=\frac{177}{7}$
8. $a_8=27$

The 6 arithmetic means inserted between 15 and 27 are: $$\frac{117}{7},\frac{129}{7},\frac{141}{7},\frac{153}{7},\frac{165}{7},\frac{177}{7}$$

Converting these fractions to decimal form (optional): $$16.71,18.43,20.14,21.86,23.57,25.29$$

Final Answer: The six arithmetic means between 15 and 27 are: $$\boxed{\frac{117}{7},\frac{129}{7},\frac{141}{7},\frac{153}{7},\frac{165}{7},\frac{177}{7}}$$

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b) Determine the common ratio of a geometric series whose first term and the fifth term are 2 and 512 respectively.

Solution:

A geometric series follows the form: $$a,ar,a{r}^2,a{r}^3,\dots$$ where:

• $a$ = first term = 2
• $r$ = common ratio (to be found)
• $n$-th term = $a{r}^{n-1}$

Given:

• First term ($a_1$) = 2
• Fifth term ($a_5$) = 512

The formula for the $n$-th term of a geometric sequence is: $$a_n=a\cdot r^{n-1}$$

For the fifth term ($n=5$): $$a_5=a\cdot r^4$$ $$512=2\cdot r^4$$ $$\frac{512}{2}=r^4$$ $$256=r^4$$

To find $r$, take the fourth root of both sides: $$r=\sqrt[4]{256}$$ $$r=4$$ (since $4^4=256$)

Verification: Let’s verify by calculating the fifth term using $r=4$: $$a_5=2\cdot 4^4=2\cdot 256=512$$ This matches the given fifth term.

Final Answer: The common ratio of the geometric series is: $$\boxed{4}$$