Connecting compound interest and the concept of limits in maths

1. A basic idea
- Ashok
Govindarajan
20-12-2017 Technology sharing series 1

2. Contents
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• 2 basic formulae
• Simple Interest
• Compound Interest
• Compound interest computation
• The fundamental idea
• Acknowledgements and References

3. 2 basic formulae
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1 +
1
𝑛
𝑛
n =1 : result is 1
n=2 : result is 9/4
n=3 : result is 64/27
.
.
.
.
n = 100 : result is (1.01)^100
= 2.7048; small number in-spite
of n= 100
1 + 𝑛 𝑛
n =1 : result is 1
n=2 : result is 9
n=3 : result is 64
.
.
.
.
n = 100 : result is (101)^100
= large number

4. Simple Interest – example and calculation
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Jan 1st /2015 Jan 1st /2016 Jan 1st /2017
Amount in
account is A =
P+ I, where P is
the principal
and I is the
interest
Re.1;
Principal is
Re.1, amount
is Re.1;
Interest yet to
start
accumulating,
hence 0.
Re.2;
Principal is
Re.1, Amount
is Re.2; Total
Interest is Re.1
Re.3;
Principal is
Re.1, Amount
is Re.3; Total
Interest is Re.2
Consider the case of simple interest offered by a Bank. The bank is a very
generous and offers 100% interest . So, the below table depicts the amount of
money present in the account over 3 years
Takeaway : The principal never changes in simple interest, at the start of the second or
the third year. It is always Re. 1! Interest is always added at the start of the new year.
Time series is : 1,2,3,4,5,6……

5. Compound Interest – example and
calculation
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Jan 1st /2015 Jan 1st /2016 Jan 1st /2017
Amount in
account
Re.1;
Principal is
Re.1, amount
is Re.1;
Interest yet to
start
accumulating,
hence 0.
Re.2;
Principal is
Re.2, Amount
is Re.2; Total
Interest is Re.1
Re.4;
Principal is
Re.4, Amount
is Re.4; Total
Interest is Re.3
Consider the case of compound interest offered by the same bank. The bank is a
very generous and offers 100% interest. So, the below table depicts the amount
of money present in the account over 3 years
Takeaway : The principal is updated in compound interest, at the start of the second and
the third year. Principal becomes the amount at the start of every new year.
The interest is added once a year as in the case of simple interest. Time series is
1,2,4,8,16, 32…..

6. Compound Interest – example and
calculation, continued
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The bank is compounding interest annually……Can it compound once every six
months?
Once every 3 months?
Once every month?
Once every day?
Once every hour?
Once every second, milli second, micro second and so on…..
Let us look at an example :
In the time series for the compound interest, let us magnify on the portion
between 1 and 2, that is the first year.

7. Compound Interest – example and calculation,
continued
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• Formula to compute compound interest is : A = P*(1+ {r/100}) ^ n
• r = 100 % for n = 1; so r = 50% for n = 2 and so on. r = 100/n
• n = 1 is 1 year, n = 2 is 6 months, n = 12 is 1 month
• So, in this context the above formula reduces to  1*(1+{1/n}) ^ n
• Once every year [n = 1 in the above formula]
• 1 --> 2.0000000000000000000000000000000
• Once every month [n = 12 in the above formula]
• 12 --> 2.6130352902246781602995330443549
• Once every hour [n = 365*24 in the above formula]
• 365*24 --> 2.7181266916204521189161380653965
• Once every second [n = 365*24*60 in the above formula]
• 365*24*60*60 -->2.7182817853609708212635582662979
• Once every millisecond, micro, nano
 WHERE ARE WE HEADING? ANY IDEAS/GUESSES?
 WILL WE GET INFINITE AMOUNT OF MONEY OR FINITE AMOUNT OF MONEY?
 Shall we take a look at slide 3, again?

8. The fundamental idea
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 The two formulae in slide 3 represents 2 cases : As “n” increases, the value of the formula
decreases and in the other case the value increases.
 Compound interest computation refers to the case where it decreases with
increase in “n “
 This idea is one of the basis for introducing limits in class 12 state board syllabus. An example
formula for limits, which is relevant here is:

 Coming back to our case, the value will converge to “e” which is approx. equal to
2.71828, based on the above formula
 So, even if compouded “continously” in time, the bank cannot offer more than 171.828 %
interest. There is a “LIMIT”
 Understanding LIMITS could facilitate the better understanding of Calculus. It is one of the
building blocks
lim
𝑛→∞
1 +
1
𝑛
𝑛
= 𝑒
Limits
As n -> infinity, 1/n -> 0

9. Acknowledgements and References
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• Dr Gilbert Strang’s lectures on “Highlights of calculus”, available on Youtube.

10. 20-12-2017 Technology sharing series 10
Thank You