What is the value of 5+45+45^2+45^3+45^4+45^5?

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

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

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.

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.

5(1+4+20+100+500+2500)
5* (3125)
5^6

Answer A

5+4.5+4.5^2+4.5^3+4.5^4+4.5^5

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

= 5 + 5^2 - 5 + 5^3 - 5^2 + 5^4 - 5^3 + 5^5 - 5^4 + 5^6 - 5^5

= 5^6 (A)

An easier way to solve this question in 45 seconds:

Start writing the equation like this:

\(5+(5-1)5+(5-1)5^2+......\)

(All 4 should be written as 5-1 since we need answer in power of 5)

Once you write it all and open brackets, you will see everything cancelling off other than \(5^6\)