Struggling with the concept of Arithmetic/Geometric Progression help

[Carvanara]

Hi all, I'm new to this website, really pleased to have found such a great website to satisfy my mathematical and scientific queries and doubts! anyway, I am currently revising for major tests.. and I can't understand/solve questions regarding Arithmetic/Geometric Progression (if you know what this topic means). Do help if possible :)

Homework Statement

so here are a few of those head crippling questions:

1) The sum of the first 100 terms of an arithmetic progression is 15050; the first, third and eleventh terms of this progression are three consecutive terms of a geometric progression. Find the first term, a and the non-zero common difference, d, of the arithmetic progression.

2) At the beginning of the year, George deposited $100,000 with a bank that pays 10% interest per annum at the end of each year. After the interest is credited, he immediately withdraws $12,000. Likewise, George will again withdraw $12,000 at the end of each subsequent year, immediately after the bank's interest has been credited. After his n-th withdrawal, he noticed, for the first time, that his bank account balance falls below $20,000. Find n.

Homework Equations

The Attempt at a Solution

Right, my attempts at solving this problem are entirely or mostly based on trial-and-error, and though I have gotten the answers through this tedious method.. let's just say that it is highly doubtful that it will work during the time-limited tests. What I need are concise steps that will enable me to solve these problems methodically.

Thanks for your muchly appreciated help!

 

[konthelion]

For starters, what's the formula for arithmetic progression/sequence?

 

[Defennder]

You'll need the formulae for sum of nth term for both arithmetic and geometric progressions, as well as the expression for the nth term.

 

[Carvanara]

ok I will edit it in :p

 

[Carvanara]

never mind, I can't figure how to edit post..

let the 1st term of a sequence be a
let the common difference in an arithmetic progression be d
let the common ratio in a geometric progression be r

nth term in an arithmetic progression: a + (n-1)(d)
Sum of n terms in an arithmetic progression: (n/2)(a + nth term)

nth term in a geometric progression: ar^(n-1)
Sum of n terms in a geometric progression: a x [(r^n)-1] / (r-1) OR a x [1-(r^n)] / (1-r)

 

[Defennder]

Ok, but how would you use these formulae in solving the questions. Start by expressing what is given in the problems in terms of these formulae. When you're done, you should have, for qn 1, a few equations which you can then solve simultaneously. Just remember that for a geometric progression, \frac{T_n}{T_{n-1}} = r.

For qn 2, observe the following:
Denote the amt deposited by "a", and amount withdrawn by "d".
End of year:

1: 1.1a - d
2: 1.1(1.1a - d) - d
3: 1.1(1.1(1.1a - d) - d) - d = 1.1^3a - 1.1^2d - 1.1d - d

Can you spot a pattern emerging here? Try to write an expression for the amount of money left in his account by the end of year n. In this case the series isn't increasing but decreasing. Now you have to apply the formulae you quoted above to solve this.

P.S. You are only allowed to edit your posts only for 30 min period after posting it.