M09-31

Question

What is the value of  5+4*5+4*5^2+4*5^3+4*5^4+4*5^5 ?

A.  5^6 
B.  5^7 
C.  5^8 
D.  5^9 
E.  5^{10}

Bunuel · Accepted Answer

Official Solution:

What is the value of  5+4*5+4*5^2+4*5^3+4*5^4+4*5^5 ?

A.  5^6 
B.  5^7 
C.  5^8 
D.  5^9 
E.  5^{10}

This question can be solved in several ways:
 
 Traditional approach : 
 
 5+4*5+4*5^2+4*5^3+4*5^4+4*5^5=5+4(5+5^2+5^3+5^4+5^5) . Note that we have the sum of a geometric progression in brackets with the first term equal to 5 and the common ratio also equal to 5. The sum of the first  n  terms of a geometric progression is given by:  	ext{sum}=\frac{b*(r^{n}-1)}{r-1} , where  b  is the first term,  n  is the number of terms, and  r  is a common ratio  
e 1 .
 
For our equation:  5+4(5+5^2+5^3+5^4+5^5)=5+4(\frac{5(5^5-1)}{5-1})=5^6 .
 
 30 sec approach based on answer choices: 
 
We have the sum of 6 terms. Now, if all terms were equal to the largest term  4*5^5 , we would have:  	ext{sum}=6*(4*5^5)=24*5^5\approx 5^2*5^5 \approx 5^7 , so the actual sum must be less than  5^7 , thus the answer must be A:  5^6 .

Answer: A

samusa · Answer

I used the following method, since I did not the formula for such a series-

5+4*5+4*5^2+4*5^3+4*5^4+4*5^5 (  WRITE 4 = 5-1  )

= 5 + ( 5-1) 5 + ( 5-1) 5^2 + (5 -1) 5 ^3 + (5 -1) 5 ^4 + (5 -1) 5 ^5 (  All the terms cancel out except 5^6  )

= 5^6

HarisinghKhedar · Answer

Good question.

The minute I saw the question, I started tackling terms in pair:
(1)5 + (4)5 = (5)5 -> 5^2
(1)5^2 + (4)5^2 = (5)5^2 -> 5^3

At end of computation, you will find 5^6.

Thanks.

Posted from my mobile device

mmagyar · Answer

This is the best solution (and Kudos to HarisinghKhedar for mentioning it). I just wanted to expand it out a little for future readers:

5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5  
  5(1+4)+4∗5^2+4∗5^3+4∗5^4+4∗5^5  
  5(5)+4∗5^2+4∗5^3+4∗5^4+4∗5^5  
  5^2+4∗5^2+4∗5^3+4∗5^4+4∗5^5  
  5^2(1+4)+4∗5^3+4∗5^4+4∗5^5  
  5^2(5)+4∗5^3+4∗5^4+4∗5^5  
  5^3+4∗5^3+4∗5^4+4∗5^5  
  5^3(1+4)+4∗5^4+4∗5^5  
  5^3(5)+4∗5^4+4∗5^5  
  5^4+4∗5^4+4∗5^5  
  5^4(1+4)+4∗5^5  
  5^4(5)+4∗5^5  
  5^5+4∗5^5  
  5^5(1+4)  
  5^5(5)  
  5^6

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Ronal_Khara · Answer

I think this is a  high-quality  question and I agree with explanation. Amazing Explanation! Beautifully presented with ease of understanding and also with a better way to solve it quickly

krishkohal · Answer