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The number of elements in the set {n ∈ {1, 2, 3, ............., 100} | (11)n > (10)n + (9)n} is ________. (in integers)
Correct answer is '96'. Can you explain this answer?
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The number of elements in the set {n∈{1, 2, 3, ............., 100...
11n > 10n + 9n
⇒ 11n - 9n > 10n
⇒ (10 + 1)n - (10 - 1)n > 10n
⇒ {nC1 . 10n-1 + nC310n-0 + nC510n-5 + .....} > 10n
⇒ 2n . 10n-1 + 2 {nC310n-3 + nC510n-5 + .......} > 10n ... (1)
For n = 6, 7, 8, .... 100
⇒ 2n10n-1 + 2 {nC310n-3 + nC510n-5 + ......} > 10n
⇒ 11n - 9n > 10n For n = 5, 6, 7, .... 100
For n = 4, inequality (1) is not satisfied.
⇒ Inequality does not hold good for N = 1, 2, 3, 4.
So, required number of elements = 96
⇒ 11n - 9n > 10n
⇒ (10 + 1)n - (10 - 1)n > 10n
⇒ {nC1 . 10n-1 + nC310n-0 + nC510n-5 + .....} > 10n
⇒ 2n . 10n-1 + 2 {nC310n-3 + nC510n-5 + .......} > 10n ... (1)
For n = 6, 7, 8, .... 100
⇒ 2n10n-1 + 2 {nC310n-3 + nC510n-5 + ......} > 10n
⇒ 11n - 9n > 10n For n = 5, 6, 7, .... 100
For n = 4, inequality (1) is not satisfied.
⇒ Inequality does not hold good for N = 1, 2, 3, 4.
So, required number of elements = 96
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Community Answer
The number of elements in the set {n∈{1, 2, 3, ............., 100...
Understanding the Inequality
To solve the inequality \( (11)^n > (10)^n + (9)^n \), we first analyze the expression for different values of \( n \).
Step-by-step Analysis
- For \( n = 1 \):
- \( 11^1 = 11 \)
- \( 10^1 + 9^1 = 10 + 9 = 19 \)
- Here, \( 11 < 19="" />
- For \( n = 2 \):
- \( 11^2 = 121 \)
- \( 10^2 + 9^2 = 100 + 81 = 181 \)
- Here, \( 121 < 181="" />
- For \( n = 3 \):
- \( 11^3 = 1331 \)
- \( 10^3 + 9^3 = 1000 + 729 = 1729 \)
- Here, \( 1331 < 1729="" />
- For \( n = 4 \):
- \( 11^4 = 14641 \)
- \( 10^4 + 9^4 = 10000 + 6561 = 16561 \)
- Here, \( 14641 < 16561="" />
- For \( n = 5 \):
- \( 11^5 = 161051 \)
- \( 10^5 + 9^5 = 100000 + 59049 = 159049 \)
- Here, \( 161051 > 159049 \).
Finding the Range of Valid \( n \)
From our calculations, we find that \( n = 1, 2, 3, 4 \) do not satisfy the inequality, but \( n = 5 \) and higher do.
Total Valid Elements
- The valid values of \( n \) start from \( 5 \) to \( 100 \).
- Thus, the total number of elements is:
\[
100 - 5 + 1 = 96
\]
Conclusion
Therefore, the number of elements in the set \( \{ n \in \{ 1, 2, 3, \ldots, 100 \} | (11)^n > (10)^n + (9)^n \} \) is **96**.
To solve the inequality \( (11)^n > (10)^n + (9)^n \), we first analyze the expression for different values of \( n \).
Step-by-step Analysis
- For \( n = 1 \):
- \( 11^1 = 11 \)
- \( 10^1 + 9^1 = 10 + 9 = 19 \)
- Here, \( 11 < 19="" />
- For \( n = 2 \):
- \( 11^2 = 121 \)
- \( 10^2 + 9^2 = 100 + 81 = 181 \)
- Here, \( 121 < 181="" />
- For \( n = 3 \):
- \( 11^3 = 1331 \)
- \( 10^3 + 9^3 = 1000 + 729 = 1729 \)
- Here, \( 1331 < 1729="" />
- For \( n = 4 \):
- \( 11^4 = 14641 \)
- \( 10^4 + 9^4 = 10000 + 6561 = 16561 \)
- Here, \( 14641 < 16561="" />
- For \( n = 5 \):
- \( 11^5 = 161051 \)
- \( 10^5 + 9^5 = 100000 + 59049 = 159049 \)
- Here, \( 161051 > 159049 \).
Finding the Range of Valid \( n \)
From our calculations, we find that \( n = 1, 2, 3, 4 \) do not satisfy the inequality, but \( n = 5 \) and higher do.
Total Valid Elements
- The valid values of \( n \) start from \( 5 \) to \( 100 \).
- Thus, the total number of elements is:
\[
100 - 5 + 1 = 96
\]
Conclusion
Therefore, the number of elements in the set \( \{ n \in \{ 1, 2, 3, \ldots, 100 \} | (11)^n > (10)^n + (9)^n \} \) is **96**.
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The number of elements in the set {n∈{1, 2, 3, ............., 100} | (11)n> (10)n+ (9)n} is ________. (in integers)Correct answer is '96'. Can you explain this answer?
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