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Sequence and Series Questions

Geometric Series.

We use the following formula

Sn=a1(1−rn)/1- r   where r ≠ 1

Example. Let’s suppose we have a chess board. A chess board has total of 64 squares (8 * 8). IF we add rice grains on first square, two rice grains on the second square, four grains on the third square and eight grains on the fourth, the sequence continues until all the squares are filled with certain amount of grains. Now find the total number of grains required to fill the chess board squares with the given sequence.

Solution. To find the sum of grains we use Geometric progression formula given above. In the given scenario, a1= 1, r = 2 (double the amount when we move to next square), n = 64 (total number of squares).

Putting values in the formula = 1 (1 – 264)/1 – 2

= 264 – 1

=  18,446,744,073,709,551,615

 

A geometric series is a series of numbers in such a manner that the numbers after the first is obtained by multiplying the previous number by a constant number. To find the sum of a finite geometric series the above formula is used.

In this scenario, the number of grains increases by 50%, that is we have to multiple the next number with a constant number i.e 2 until the whole 64th square.We do not multiply 54 by two because there is no next number and we multiply 2 until 63 only.

1.1. Series Question

Formula.  Un = U1 Un-1

Example. With the given sequence 12,6,3,1.5,0.75… Find the 10th number in sequence and the general Un term.

Solution. By using the formula above.

Un = 10. (½) n-1

Or, Un = 23*2-1(n-1)

Un = 23 * 2n + 1

Un = 23+1-n

Un = 24-n

To find the 10th number in sequence.

U10 = 24-10

= 0.015

 

  1. Future value of Compound Interest

Formula. FV = PV (1 + r/n) tn

Example. If you deposit an amount of $5000 into a saving account that has 7% annual interest, compounded monthly. You keep your money in the bank for 10 years and then withdraw it. And, your friend deposit the same amount at the same bank but withdraw the amount after 8 years.

Calculate the total amount you will and your friend will have after you both withdraw.

Solution.

Using the above formula, here PV is $5000, r = 7% or 0.07, n = 12 (months in a year), t = 10(yours) and 8 (for your friend).

If we put the values

FV = 5000 (1 + 0.07/12) 12 * 10

= 10,044.31

After 10 years, you will have $10,044.31 in your bank account.

FV = 5000(1 + 0.07/12) 12*8

= 8,736.2524

While your friend will have $8,736.25.

The thing is, compound interest unleashes its power with time. The more time passes, the stronger it becomes. “Compound interest is a magic”

Even tough, you and your friend has save the same amount, given the same interest rate, but your friend withdrew the amount just 2 years before you and look at the difference of amount he has lost.

  1. Future Value of Annuity

Formula.  FV = A ((1 + r)n– 1)/ r

Where; r is interest rate

A is the annuity amount and n is the number of periods

Example.  You are paying a student loan of $1000 each month, for the next 12 months. The monthly interest rate in 13%. What is the future value of the payments that you are making?

Solution. To know the future value of your payments, use the above formula.

FV = 1000((1 + 0.13)12 – 1) / 0.13

$11,1000

After 12 months, you’d have paid total amount of $11,100 in your student loan. Because each month the amount of interest passes an incremental increase. First month you pay 1,130, the next month you pay 1,130*1.13 = 1276.9. The month after that 1442.89 (1276.9 * 1.13).

Formula.  A = P ((r (1 + r)n – 1 ) / ((1 + r)n – 1) )

P = loan payment

A = Periodic repayment amount.

n = number of repayments.

i = interest rate.

Example. You borrowed $10,000 at a rate of 6%. You want to repay it in five equal instalments over five years, with the first repayment one year after you out the loan. How much should each repayment be?

Solution.  With equal amount of payments each year, you will have to

A = 10,000 * 0.06(1.06)5 / ((1.06)5 – 1)

= 2373.96

You will pay $2373.96 each year for five years.

Even though you pay the same amount each year. But, with each year you pay less interest because the total amount of loan decreases. In the first year you pay an interest of $600. In the second year, you pay interest of $493.56 because the interest is charged on $8226.04 (10,000 – 2373.96). For the third year you pay an interest of 380.74.

 

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