Which is greater 105^19 or 100^20?

Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.

Dear fellowmembers,

I have come across a question that I am not able to either solve it or find a suitable explanation; the question is: -

Q). Which is greater 105^19 or 100^20?

Please help!
Thanks in advance.

Let me try :


1st expression : \(105^{19}\) \(= (100 *1.05)^{19}\) \(= 100^{19}*1.05^{19}\) --- [ as \((a*b)^n = (a^n)*(b^n)\) ]

2nd expression : \(100^{20}\) \(= 100^{19} * 100^1\) \(= 100^{19}*100\) --- [ as \(a^n = (a^1)*(a^{n-1})\) ]

now as \(100^{19}\) is common,

Compare \(1.05^{19}\) with \(100\).

as 1.05 is slightly greater than 1

surely, \(1.05^{19} < 100\)


thus

\(105^{19} < 100^{20}\)

Let me know if this worked for you.

Could you change the main subject as 'Which is greater 105^19 or 100^20?' So, It may be helpful for others too, in case they need to dive into the problem.

goutamread has given you a great approach. I would use a similar one though I just try to simplify first. It helps you think faster in some cases.

\(105^{19}\) and \(100^{20}\) have \(5^{19}\) common. Cancel out the common \(5^{19}\) and you are left with

\(105^{19} = 5^{19}*21^{19}\)
and \(100^{20} = 5^{19}*20^{19}*100\)

Now compare \(21^{19}\) and \(20^{19}*100\)

It is easy to make some sense out of \(20^{19}*100\). It is 100 times \(20^{19}\) i.e. you can get this expression if you add 100 terms of \(20^{19}\) together

\(20^{19}*100 = 20^{19} + 20^{19} + 20^{19}\) ... (100 terms) ..............(I)

To compare, we can use binomial for \(21^{19}\). If you do not know how to use binomial, check out this link:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/05 ... ek-in-you/

\(21^{19} = (20 + 1)^{19}\)

Let's try to expand \((20 + 1)^{19}\) using binomial. We will get 19+1 = 20 terms when we expand.

\((20 + 1)^{19} = 20^{19} + 19*20^{18} + 19* 18/2 * 20^{17} + ... + 1^{19}\) ............... (II)

Notice that this expression has 20 terms and none of the terms will be greater than \(20^{19}\).

Compare (I) with (II).
(I) has 100 terms, all of them \(20^{19}\)
(II) has 20 terms, all of them equal to or less than \(20^{19}\).

Hence, (I) will be greater than (II). Or we can say that \(100^{20}\) will be greater than \(105^{19}\).

Thanks a lot goutamread! This one is really better. But, I want to know that what made you to split 105 into 100*1.05. This is some thing that is really very important to know because I believe that these are the tactics that one has to think of while attacking a question.