Struggling with the concept of Arithmetic/Geometric Progression help

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In summary, the conversation discusses the topics of Arithmetic and Geometric Progression and how to solve problems related to them. The first question involves finding the first term and common difference in an arithmetic progression when given the sum of the first 100 terms and the fact that the first, third, and eleventh terms are consecutive terms in a geometric progression. The second question involves finding the number of times a person can withdraw money from a bank account before the balance falls below a certain amount, using formulas for arithmetic and geometric progressions. The conversation also includes a discussion on how to use these formulas to solve the problems systematically.
  • #1
Carvanara
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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!
 
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  • #2
For starters, what's the formula for arithmetic progression/sequence?
 
  • #3
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.
 
  • #4
ok I will edit it in :p
 
  • #5
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)
 
  • #6
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, [tex]\frac{T_n}{T_{n-1}} = r[/tex].

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Denote the amt deposited by "a", and amount withdrawn by "d".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

1: [tex]1.1a - d[/tex]
2: [tex]1.1(1.1a - d) - d[/tex]
3: [tex]1.1(1.1(1.1a - d) - d) - d = 1.1^3a - 1.1^2d - 1.1d - d[/tex]

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.
 

1. What is the difference between arithmetic and geometric progression?

Arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is always the same. Geometric progression is a sequence of numbers in which the ratio between any two consecutive terms is always the same.

2. How can I determine the next term in an arithmetic progression?

To determine the next term in an arithmetic progression, you need to first find the common difference between the terms. Then, add this difference to the last term to find the next term in the sequence.

3. What is the formula for finding the sum of an arithmetic or geometric progression?

The formula for finding the sum of an arithmetic progression is: Sn = (n/2)(2a + (n-1)d), where a is the first term, d is the common difference, and n is the number of terms. The formula for finding the sum of a geometric progression is: Sn = a((1-r^n)/(1-r)), where a is the first term, r is the common ratio, and n is the number of terms.

4. How can I identify if a sequence is an arithmetic or geometric progression?

To identify if a sequence is an arithmetic or geometric progression, you can look for a pattern in the differences between consecutive terms. If the differences are the same, it is an arithmetic progression. If the ratios between consecutive terms are the same, it is a geometric progression.

5. Are there any real-world applications of arithmetic and geometric progression?

Yes, arithmetic and geometric progressions have many real-world applications. For example, they can be used to calculate compound interest in finance, determine population growth in biology, and analyze data trends in statistics.

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