I'm having trouble getting the number of terms for the sum of this geometric progression.

Question

This question has been making me mad all day! It's in a advanced maths text book and my teacher asked us to do it for homework.
Here's the question:

How many terms of the sequence 4, 3, 2.25, ... can you add before the sum exceeds 12?

Here's my working out:


The answer I got is n=-2 and it's incorrect. I checked the answer for this question at back of text book and it was n=4. I tried and tried but still got n=-2. Please help!

Answer 1

Hint. From the line $$ 1-\left(\frac34 \right)^n>\frac34 $$ you get $$ \left(\frac34 \right)^n<\frac14 $$ giving $$ n>\frac{\log(1/4)}{\log(3/4)}=4.8\ldots $$ that is $$n=5.$$

Answer 2

There is an error in the second row of the second column. The inequality should be $$\left(\frac{3}{4}\right)^n <0.25$$

Edit: to be fair there are some other errors after that point, but that's the first one. Remember that dividing for a negative number changes the direction of the inequality

Answer 3

Try subtracting 1 from both sides instead of adding (second column, second row)