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

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What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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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

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Originally posted by happyface101 on 16 Jan 2016, 20:03.
Last edited by Bunuel on 22 Jan 2016, 09:13, edited 2 times in total.
Updated the topic.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 16 Jan 2016, 20:08
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Everyone please see below for the official answer. The only piece I don't quite understand is how 5+4(5(5^5−1)/(5−1)) became 5^6. Can someone please explain the in between steps to me? Thank you!


Official Answer

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 geometric progression in brackets with first term equal to 5 and common ratio also equal to 5. The sum of the first n terms of geometric progression is given by: sum=b∗(r^n−1)/(r−1), where b is the first term, n the number of terms and r is a common ratio ≠1.

So in our case: 5+4(5+5^2+5^3+5^4+5^5)=5+4(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: sum=6∗(4∗5^5)=24∗5^5≈5^2∗5^5≈57, so the actual sum must be less than 5^7, thus the answer must be A: 5^6.

Answer: A
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 16 Jan 2016, 20:13
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happyface101 wrote:
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


Follow posting guidelines especially on how to write the topic title.

As for the question,

\(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5 = 5+4∗5(1+5+5^2+5^3+5^4)\)

---> Realize that (1+5+5^2+5^3+5^4) is a geometric progression series with first term (a) = 1 and ratio, r = 5.

Sum of a G.P. with first term (a) and ratio, r (>1) = \(\frac{a(r^n-1)}{(r-1)}\), where n = number of terms in the series = 5.

Thus 5+4∗5(1+5+5^2+5^3+5^4) = 5+4*5*[S] where S= \(\frac{a(r^n-1)}{(r-1)}\) = \(\frac{1(5^5-1)}{(5-1)}\) = \(\frac{5^5-1}{4}\)

---> \(5+4*5*[S] = 5+4*5*(\frac{5^5-1}{4}) = 5+5^6-5 = 5^6\).

Hence A is the correct answer.

Hope this helps.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 16 Jan 2016, 20:18
happyface101 wrote:
Everyone please see below for the official answer. The only piece I don't quite understand is how 5+4(5(5^5−1)/(5−1)) became 5^6. Can someone please explain the in between steps to me? Thank you!



As mentioned in my post above, once you get S = \(\frac{5^5-1}{4}\), you are now left with \(4*5*S = 4*5*\frac{5^5-1}{4}\)

---> You get, \(5(5^5-1) = 5*5^5-5 = 5^6-5\)

Finally, \(5+4*5*S\)becomes \(5+5^6-5 = 5^6\)

Hope this helps.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 16 Jan 2016, 20:26
1
happyface101 wrote:
Everyone please see below for the official answer. The only piece I don't quite understand is how 5+4(5(5^5−1)/(5−1)) became 5^6. Can someone please explain the in between steps to me? Thank you!


Official Answer

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 geometric progression in brackets with first term equal to 5 and common ratio also equal to 5. The sum of the first n terms of geometric progression is given by: sum=b∗(r^n−1)/(r−1), where b is the first term, n the number of terms and r is a common ratio ≠1.

So in our case: 5+4(5+5^2+5^3+5^4+5^5)=5+4(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: sum=6∗(4∗5^5)=24∗5^5≈5^2∗5^5≈57, so the actual sum must be less than 5^7, thus the answer must be A: 5^6.

Answer: A


hi,
the simplest would be..
\(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5\)..
keep adding from left..
\((5+4∗5)+4∗5^2+4∗5^3+4∗5^4+4∗5^5\)...
\((5*5+4∗5^2)+4∗5^3+4∗5^4+4∗5^5\)....
\((5^3+4∗5^3)+4∗5^4+4∗5^5\)..
\((5^4+4∗5^4)+4∗5^5\)..
\(5^5+4∗5^5\)...
\(5^6\)..

the way "5+4(5(5^5−1)/(5−1))" would take some time to get to 5^6...
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 16 Jan 2016, 20:36
Engr2012 wrote:
happyface101 wrote:
Everyone please see below for the official answer. The only piece I don't quite understand is how 5+4(5(5^5−1)/(5−1)) became 5^6. Can someone please explain the in between steps to me? Thank you!



As mentioned in my post above, once you get S = \(\frac{5^5-1}{4}\), you are now left with \(4*5*S = 4*5*\frac{5^5-1}{4}\)

---> You get, \(5(5^5-1) = 5*5^5-5 = 5^6-5\)

Finally, \(5+4*5*S\)becomes \(5+5^6-5 = 5^6\)

Hope this helps.


Wow must have had a brain freeze :( thank you! +1
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 20 Jan 2016, 15:21
i totally don't understand this at all... how did you guys get from 5^6-5 to the next step?
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 20 Jan 2016, 15:34
nycgirl212 wrote:
i totally don't understand this at all... how did you guys get from 5^6-5 to the next step?


Once you assume that \(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5=5+4∗5*(1+5+5^2+5^3+5^4)\), such that \(S = (1+5+5^2+5^3+5^4)\).

The original question is thus = \(5+4*5*S\) ....(1)

As mentioned above, series 'S' is a geometric progression that will have a sum = \(S = \frac{5^5-1}{4}\)

Thus your original series from (1) above = \(5+4∗5∗S = 5+4*5*\frac{5^5-1}{4}\), the portion after the '+' sign becomes = \(5*(5^5-1)\)

Finally, \(5+4∗5∗S = 5+5*(5^5-1) = 5+5*5^5-5*1 = 5+5^6-5 = 5^6\) ...(employed the rule that \(a^b*a^c = a^{b+c}\))

Hope this helps.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 20 Jan 2016, 15:44
Engr2012 wrote:
nycgirl212 wrote:
i totally don't understand this at all... how did you guys get from 5^6-5 to the next step?


Once you assume that \(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5=5+4∗5*(1+5+5^2+5^3+5^4)\), such that \(S = (1+5+5^2+5^3+5^4)\).

The original question is thus = \(5+4*5*S\) ....(1)

As mentioned above, series 'S' is a geometric progression that will have a sum = \(S = \frac{5^5-1}{4}\)

Thus your original series from (1) above = \(5+4∗5∗S = 5+4*5*\frac{5^5-1}{4}\), the portion after the '+' sign becomes = \(5*(5^5-1)\)

Finally, \(5+4∗5∗S = 5+5*(5^5-1) = 5+5*5^5-5*1 = 5+5^6-5 = 5^6\) ...(employed the rule that \(a^b*a^c = a^{b+c}\))

Hope this helps.


that helps, thank you... confused on this part... sum = \(S = \frac{5^5-1}{4}\).... is this a set formula?
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 20 Jan 2016, 15:46
nycgirl212 wrote:
Engr2012 wrote:
nycgirl212 wrote:
i totally don't understand this at all... how did you guys get from 5^6-5 to the next step?


Once you assume that \(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5=5+4∗5*(1+5+5^2+5^3+5^4)\), such that \(S = (1+5+5^2+5^3+5^4)\).

The original question is thus = \(5+4*5*S\) ....(1)

As mentioned above, series 'S' is a geometric progression that will have a sum = \(S = \frac{5^5-1}{4}\)

Thus your original series from (1) above = \(5+4∗5∗S = 5+4*5*\frac{5^5-1}{4}\), the portion after the '+' sign becomes = \(5*(5^5-1)\)

Finally, \(5+4∗5∗S = 5+5*(5^5-1) = 5+5*5^5-5*1 = 5+5^6-5 = 5^6\) ...(employed the rule that \(a^b*a^c = a^{b+c}\))

Hope this helps.


that helps, thank you... confused on this part... sum = \(S = \frac{5^5-1}{4}\).... is this a set formula?


Yes. It is from the formula for calculation of sum of a geometric progression. Refer to what-is-the-value-of-211922.html#p1631285 for more details.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 22 Jan 2016, 09:02
Engr2012 wrote:
happyface101 wrote:
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


Follow posting guidelines especially on how to write the topic title.

As for the question,

\(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5 = 5+4∗5(1+5+5^2+5^3+5^4)\)

---> Realize that (1+5+5^2+5^3+5^4) is a geometric progression series with first term (a) = 1 and ratio, r = 5.

Sum of a G.P. with first term (a) and ratio, r (>1) = \(\frac{a(r^n-1)}{(r-1)}\), where n = number of terms in the series = 5.

Thus 5+4∗5(1+5+5^2+5^3+5^4) = 5+4*5*[S] where S= \(\frac{a(r^n-1)}{(r-1)}\) = \(\frac{1(5^5-1)}{(5-1)}\) = \(\frac{5^5-1}{4}\)

---> \(5+4*5*[S] = 5+4*5*(\frac{5^5-1}{4}) = 5+5^6-5 = 5^6\).

Hence A is the correct answer.

Hope this helps.


trying to understand the formula for the geometric progression.... so the ratio of 5, which term is that?
and the number of terms in the series is that the count of the summable amounts in here? (1+5+5^2+5^3+5^4)
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5?  [#permalink]

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New post 22 Jan 2016, 09:09
nycgirl212 wrote:
Engr2012 wrote:
happyface101 wrote:
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


Follow posting guidelines especially on how to write the topic title.

As for the question,

\(5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5 = 5+4∗5(1+5+5^2+5^3+5^4)\)

---> Realize that (1+5+5^2+5^3+5^4) is a geometric progression series with first term (a) = 1 and ratio, r = 5.

Sum of a G.P. with first term (a) and ratio, r (>1) = \(\frac{a*(r^n-1)}{(r-1)}\), where n = number of terms in the series = 5.

Thus 5+4∗5(1+5+5^2+5^3+5^4) = 5+4*5*[S] where S= \(\frac{a(r^n-1)}{(r-1)}\) = \(\frac{1(5^5-1)}{(5-1)}\) = \(\frac{5^5-1}{4}\)

---> \(5+4*5*[S] = 5+4*5*(\frac{5^5-1}{4}) = 5+5^6-5 = 5^6\).

Hence A is the correct answer.

Hope this helps.


trying to understand the formula for the geometric progression.... so the ratio of 5, which term is that?
and the number of terms in the series is that the count of the summable amounts in here? (1+5+5^2+5^3+5^4)


A geometric progression is a series of the form \(a+a*r+a*r^2+a*r^3....a*r^{n-1}\), wherein a= first term of the series, r = common ratio = 2nd term/first term or 3rd term/ 2nd term or 4th term/3rd term etc.

Sum of such a geometric series =\(\frac{a*(r^n-1)}{r-1}\)

Thus for the series given to us in the question, \(1+5+5^2+5^3+5^4\) , has the first term = a=1, ratio = r= 5/1 or 5^2/5 = 5, number of terms = n=5

Thus the sum of the series \(1+5+5^2+5^3+5^4\)=\(1*(5^n-1)/(5-1)\) = \((5^5-1)/4\)

Hope this helps.
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Re: What is the value of 5+4∗5+4∗5^2+4∗5^3+4∗5^4+4∗5^5? &nbs [#permalink] 22 Jan 2016, 09:09
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