Solving Geometric Progression & Logarithmic Equations Math Questions

[I'm clever]

I'm stuck on these three maths questions.

1) In a geometric progression, the sum to infinity is four times the first term.

(i) Show that the common ratio is 3
(ii) Given that the third term is 9, find the first term.
(iii) Find the sum of the first twenty terms.


2) Solve the equation log_2 x + 2 log_2 3 = log_2(x + 5).

3) Find:

200
Σ (3n+2)
n=101

for 3) Should I subtract the series cause it doesn't start with 1?

 

[eumyang]

Show us what you have tried. (We can't just give answers here.) For #2 you'll need to use the logarithmic property
$$\log_b xy = \log_b x + \log_b y$$
... among others.

 

[I'm clever]

Show us what you have tried. (We can't just give answers here.) For #2 you'll need to use the logarithmic property
$$\log_b xy = \log_b x + \log_b y$$
... among others.

$$\log_2 x + 2\log_2 3 = \log_2 (x+5)$$

Power rule:

$$\log_2 x + \log_2 3^2 = \log_2 (x+5)$$

Addition rule:

$$\log_2 (x \times 3x^2) = \log_2 (x+5)$$

$$\log_2 3x^2 = \log_2 (x+5)$$

 

[I'm clever]

$$\log_2 x + 2\log_2 3 = \log_2 (x+5)$$

Power rule:

$$\log_2 x + \log_2 3^2 = \log_2 (x+5)$$

Addition rule:

$$\log_2 (x \times 3x^2) = \log_2 (x+5)$$

$$\log_2 3x^2 = \log_2 (x+5)$$

Is this right?

 

[eumyang]

$$\log_2 x + 2\log_2 3 = \log_2 (x+5)$$

Power rule:

$$\log_2 x + \log_2 3^2 = \log_2 (x+5)$$

Addition rule:

$$\log_2 (x \times 3x^2) = \log_2 (x+5)$$

Where did that second "x" come from? It should be
$$\log_2 (x \times 3^2) = \log_2 (x+5)$$
or
$$\log_2 (9x) = \log_2 (x+5)$$

After this, use the property:
if log_b x = log_b y, then x = y
... and solve for x.

 

[eumyang]

I'm stuck on these three maths questions.

1) In a geometric progression, the sum to infinity is four times the first term.

(i) Show that the common ratio is 3

Are you sure you copied the problem right? My understanding is that unless r < 1 the series won't converge. If the sum to infinity is four times the first term, then I'm getting 3/4 as the common ratio, not 3.