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If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9

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If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 11 Mar 2020, 23:51
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A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

27% (01:55) correct 73% (02:17) wrong based on 41 sessions

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If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 12 Mar 2020, 01:09
1
1
\(S=k*10^9= 10^9+2*11*10^8+3*11^2*10^7+........+10*11^9\)........(1)

\((\frac{11}{10})S= 11*10^8+2*11^2*10^7+........+9*11^9+11^{10}\)......(2)

Subtract (1) from (2)

\(\frac{S}{10} = 11^{10}- [ 10^9+11*10^8+11^2*10^7........+11^9]\)

S/10 = 11^{10}- [10^9{(11/10)^{10} - 1}/{(11/10)-1}]

\(\frac{S}{10} = 11^{10} -[10^{10} ((\frac{11}{10})^{10} - 1)]\)

\(\frac{S}{10} = 11^{10} -[11^{10} - 10^{10}]\)

\(S= 10^{11}\)

\(k*10^9= 10^{11}\)

\(k= 10^2 = 100\)
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If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post Updated on: 13 Mar 2020, 17:21
4
10^9 * { 1+ 2*(1.1)^1 + 3*(1.1)^2 +... + 10*(1.1)^9 } = 10^9 * k

Estimate lower-bound value of k:
= 10^9 * { 1+ 2*(1)^1 + 3*(1)^2 +... + 10*(1)^9 }
= 10^9 * {1+ 2+ ...+10}
= 10^9 * 55 = 10^9 * k
k > 55 ---> eliminate choices (A) 12.1 and (B) 44.1

Estimate reasonable value of k:
= 10^9 * { 1+ 2*(1)^1 + 3*(1)^2 + ... +10*(1)^9} * 1.1^6
= 10^9 * 55 * ~1.6
= 10^9 * ~88 = 10^9 * k
k is close to 100

Thus, FINAL ANSWER IS (C)

Originally posted by chondro48 on 12 Mar 2020, 01:50.
Last edited by chondro48 on 13 Mar 2020, 17:21, edited 4 times in total.
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Re: If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 12 Mar 2020, 05:40
I have calculated the whole log. It comes down to 10 to the power 11. So the answer should be C. But do they ask such questions in GMAT? If they do, there must be some way of doing it within 2 minutes which I cannot even begin to search for!
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Re: If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 12 Mar 2020, 10:03
1
If \((10)^9+2(11)^1(10)^8+3(11)^2(10)^7+…+10(11)^9=k(10)^9\), then k is equal to

A. 121/10
B. 441/100
C. 100
D. 110
E. 120
Multiplying given equation's both sides by \(10^{-9}\)
\((10)^9*10^{-9} + 2(11)^1(10)^8*10^{-9} + 3(11)^2(10)^7*10^{-9} + 4(11)^3(10)^6*10^{-9} + 5(11)^4(10)^5*10^{-9} + 6(11)^5(10)^4*10^{-9}+\)
\(+ 7(11)^6(10)^3*10^{-9} + 8(11)^7(10)^2*10^{-9} + 9(11)^8(10)^1*10^{-9} + 10(11)^9*10^{-9} = k(10)^9*10^{-9}\)

\(1 + 2*11*10^{-1} + 3(11)^2*10^{-2} + 4(11)^3*10^{-3} + 5(11)^4*10^{-4} + 6(11)^5*10^{-5} + 7(11)^6*10^{-6}+\)
\(+ 8(11)^7*10^{-7} + 9(11)^8*10^{-8} + 10(11)^9*10^{-9} = k\)

\(1 + 2*1.1^1 + 3*1.1^2 + 4*1.1^3 + 51.1^4 + 6*1.1^5 + 7*1.1^6 + 8*1.1^7 + 9*1.1^8 + 10*1.1^9 = k\)

Solving for k gives ~ 100
(There must be a better way to solve it)

Answer C.
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Re: If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 12 Mar 2020, 23:33
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1
If \((10)^9+2 (11)^{1}(10)^{8}+3 (11)^{2}(10)^{7}+…+10 (11)^{9}=k (10)^{9}\), then k is equal to

\(10^9*( 1 +2 (\frac{11}{10})+3 (\frac{11}{10})^{2}+…+10 (\frac{11}{10})^{9}=k (10)^{9}\)

\(k= 1 +2 (\frac{11}{10})+3 (\frac{11}{10})^{2}+…+10 (\frac{11}{10})^{9}\)
---------------------------------------------------------------------------------------------------------------------------------------------
--> multiply both sides by \(\frac{11}{10}\):
\(k(\frac{11}{10})= (\frac{11}{10}) + 2(\frac{11}{10})^{2} + 3(\frac{11}{10})^{3} +...+ 10(\frac{11}{10})^10\)

And subtract \(k\) from \(k(\frac{11}{10})\):

\(k(\frac{11}{10} -1)= -1 -( (\frac{11}{10}) + (\frac{11}{10})^{2} + (\frac{11}{10})^{3} +...+ (\frac{11}{10})^{9} ) + 10(\frac{11}{10})^{10}\)
---------------------------------------------------------------------------------------------------------------------------------------------
\((\frac{11}{10}) + (\frac{11}{10})^{2} + (\frac{11}{10})^{3} +...+ (\frac{11}{10})^{9}\) - geometric sequence

\(Sum = \frac{(\frac{11}{10})*(\frac{11}{10})^9 -1)}{(\frac{11}{10}-1)}= 11* (\frac{11}{10})^9 -1)\)


---------------------------------------------------------------------------------------------------------------------------------------------
\(k(\frac{11}{10} -1)= -1 - \frac{11^{10}}{10^{9}} +11 + \frac{11^{10}}{10^{9}}= 10\)

\(k (\frac{1}{10}) = 10\)
\(k =100\)

Answer (C)
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Re: If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9  [#permalink]

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New post 15 Mar 2020, 03:41
Bunuel wrote:

Competition Mode Question



If \((10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9\), then k is equal to

A. 121/10
B. 441/100
C. 100
D. 110
E. 120




Since 11 = 1.1 x 10, we can rewrite the left hand side of the equation as:

10^9 + 2(1.1 x 10)(10^8) + 3(1.1^2 x 10^2)(10^7) + … + 10(1.1^9 x 10^9) = k(10^9)

10^9 + 2(1.1)(10^9) + 3(1.1^2)(10^9) + … + 10(1.1^9 )(10^9) = k(10^9)

Dividing the equation by 10^9, we have:

1 + 2(1.1) + 3(1.1)^2 + … + 10(1.1^9) = k → Eq. 1

We need to find the value of k. Without a calculator, we can multiply the equation by 1.1 to obtain:

1.1 + 2(1.1)^2 + 3(1.1)^3 + … + 10(1.1^10) = 1.1k → Eq. 2

Subtracting Eq. 2 from Eq. 1 (by combining terms with the same powers of 1.1), we have:

1 + 1.1 + 1.1^2 + 1.1^3 + … + 1.1^9 - 10(1.1^10) = -0.1k → Eq. 3

Now we see that all the terms (except the last term) form a geometric sequence (notice that the first term 1 can be written as 1.1^0).

Recall that the sum of a finite geometric series is:

S_n = a_1 * (1 - r^n) / (1 - r)

where a_1 is the first term, n is the number of terms and r is the common ratio.

So the sum of the 10 positive terms (i.e., 1, 1.1, 1.1^2, …, 1.1^9) is:

S_10 = 1(1 - 1.1^10)/(1 - 1.1) = (1 - 1.1^10)/(-0.1)

Substituting this back into Eq. 3, we have:

(1 - 1.1^10)/(-0.1) - 10(1.1^10) = -0.1k

Multiply the above equation by -0.1, we have:

1 - 1.1^10 + 1.1^10 = 0.01k

1 = 0.01k

100 = k

Answer: C
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Re: If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9   [#permalink] 15 Mar 2020, 03:41

If (10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + … + 10(11)^9 = k(10)^9

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