I have been stuck on this problem for about an hour, can someone please help me?
in Trigonometry Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

On Gideon’s first birthday, Dr Prince deposits $1400. 4.6% as a decimal is 0.046, so on the second birthday, the original $1400 grows by the factor 1.046, becoming $1464.40. Dr Prince deposits a further $1400, making the amount at the start of the second year $2864.40.

On the third birthday $2864.40 will grow by the factor 1.046 to $2996.16.

Let’s write this down symbolically. Let the deposit be D and the rate r (so D=1400 and r=0.046).

Growth at end of 1st year=D(1+r)

Amount at beginning of 2nd year=D+D(1+r)

Growth at end of 2nd year=(D+D(1+r))(1+r)=D(1+r)+D(1+r)²

Amount at beginning of 3rd year=D+D(1+r)+D(1+r)²

Growth at end of 3rd year=(D+D(1+r)+D(1+r)²)(1+r).

So we can write this as D(1+r+(1+r)²+(1+r)³).

Fast forward to the end of the nth year, Gideon’s (n+1)th birthday, assuming that his first birthday is when he is 1 year old.

We have:

D(1+r+(1+r)²+(1+r)³+...+(1+r)ⁿ).

We can write this:

D(1+r)[1+1+r+(1+r)²+(1+r)³+...+(1+r)ⁿ⁻¹].

In the square parentheses we have a geometric progression so we can use the sum formula: S(n)=[(1+r)ⁿ-1]/[1+r-1]=[(1+r)ⁿ-1]/r.

At the end of the nth year the accumulated amount is:

D(1+r)S(n)=[D(1+r)/r][(1+r)ⁿ-1].

Let’s check this by using the given values to discover what has accumulated by the time Gideon is 3 years old, so n+1=3, n=2:

(1400×1.046/0.046)(0.094116)=$2996.16, which is what was calculated earlier.

You have not asked a specific question, but you now have a formula for calculating how much Gideon would have by the time he is, say, 18 years old.

by Top Rated User (695k points)

Related questions

Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
84,102 questions
89,037 answers
1,992 comments
6,728 users