Question

Show work and explain with formulas.

20. Find the first term in a geometric progression whose common ratio is 3 and whose 5th term is 324.

21. Which term is 1/64 in the geometric progression 64, 32, 16, ...?

22. In a geometric progression, the first term is 1875, the nth term is 48, the common ratio is 2/5. Find the value of n.​

Answer 1

20 Answer: a₁ = 4

Explanation:

\begin{lgathered}a_n=324,\ r=3,\ n=5\\\\a_n=a_1 \cdot r^{n-1}\\\\324=a_1\cdot 3^{5-1}\\\\\dfrac{324}{3^4}=a_1\\\\\dfrac{324}{81}=a_1\\\\\large\boxed{4}=a_1\end{lgathered}

a

n

=324, r=3, n=5

a

n

=a

1

⋅r

n−1

324=a

1

⋅3

5−1

3

4

324

=a

1

81

324

=a

1

4

=a

1

21 Answer: n = 13

Explanation:

\begin{lgathered}a_n=\dfrac{1}{64},\ a_1=64,\ r=\dfrac{1}{2}\\\\a_n=a_1 \cdot r^{n-1}\\\\\dfrac{1}{64}=64\cdot \bigg(\dfrac{1}{2}\bigg)^{n-1}\\\\\dfrac{1}{64\cdot 64}=\bigg(\dfrac{1}{2}\bigg)^{n-1}\\\\\dfrac{1}{2^6\cdot 2^6}=\dfrac{1}{2^{n-1}}\\\\6+6=n-1\\\\\large\boxed{13}=n\end{lgathered}

a

n

=

64

1

, a

1

=64, r=

2

1

a

n

=a

1

⋅r

n−1

64

1

=64⋅(

2

1

)

n−1

64⋅64

1

=(

2

1

)

n−1

2

6

⋅2

6

1

=

2

n−1

1

6+6=n−1

13

=n

22 Answer: n = 5

Explanation:

Answer 2

20 Answer: a₁ = 4

Explanation:

`a_n=324,\ r=3,\ n=5\\\\a_n=a_1 \cdot r^(n-1)\\\\324=a_1\cdot 3^(5-1)\\\\(324)/(3^4)=a_1\\\\(324)/(81)=a_1\\\\\large\boxed{4}=a_1`

21 Answer: n = 13

Explanation:

`a_n=(1)/(64),\ a_1=64,\ r=(1)/(2)\\\\a_n=a_1 \cdot r^(n-1)\\\\(1)/(64)=64\cdot \bigg((1)/(2)\bigg)^(n-1)\\\\(1)/(64\cdot 64)=\bigg((1)/(2)\bigg)^(n-1)\\\\(1)/(2^6\cdot 2^6)=(1)/(2^(n-1))\\\\6+6=n-1\\\\\large\boxed{13}=n`

22 Answer: n = 5

Explanation:

`a_n=48,\ a_1=1875\ r=(2)/(5)\\\\a_n=a_1 \cdot r^(n-1)\\\\48=1875\cdot \bigg((2)/(5)\bigg)^(n-1)\\\\(48)/(1875)=\bigg((2)/(5)\bigg)^(n-1)\\\\(16)/(625)=\bigg((2)/(5)\bigg)^(n-1)}\\\\\bigg((2)/(5)\bigg)^4=\bigg((2)/(5)\bigg)^(n-1)}\\\\4=n-1\\\\\large\boxed{5}=n`