Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > For the sequence 1, 7, 25, 79, 241, 727 &hell...
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For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {an} is ____________
- a)3n+1 – 2
- b)3n – 2
- c)(-3)n + 4
- d)n2 – 2
Correct answer is option 'B'. Can you explain this answer?
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For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {a...
The ratio of consecutive numbers is close to 3. Comparing these terms with the sequence of {3n} which is 3, 9, 27 …. Comparing these terms with the corresponding terms of sequence {3n} and the nth term is 2 less than the corresponding power of 3.
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For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {a...
Understanding the Sequence
The sequence provided is: 1, 7, 25, 79, 241, 727. To find a simple formula for the nth term, we can analyze the differences between consecutive terms.
Calculating Differences
- First, let's identify the differences between the terms:
- 7 - 1 = 6
- 25 - 7 = 18
- 79 - 25 = 54
- 241 - 79 = 162
- 727 - 241 = 486
- The differences are: 6, 18, 54, 162, 486.
- Now, let's calculate the second differences:
- 18 - 6 = 12
- 54 - 18 = 36
- 162 - 54 = 108
- 486 - 162 = 324
- The second differences are: 12, 36, 108, 324, which have a common ratio (3).
Identifying the Formula
The pattern in the differences suggests a geometric progression, hinting that the nth term has an exponential factor.
- Observing the terms, we can propose a formula of the form:
\[ a_n = 3^n - 2 \]
- Testing this formula with the values:
- For n=1: \( a_1 = 3^1 - 2 = 1 \)
- For n=2: \( a_2 = 3^2 - 2 = 7 \)
- For n=3: \( a_3 = 3^3 - 2 = 25 \)
- For n=4: \( a_4 = 3^4 - 2 = 79 \)
- For n=5: \( a_5 = 3^5 - 2 = 241 \)
- For n=6: \( a_6 = 3^6 - 2 = 727 \)
Conclusion
All terms match the original sequence, confirming that the correct answer is indeed:
**Option B: \( 3^n - 2 \)**
The sequence provided is: 1, 7, 25, 79, 241, 727. To find a simple formula for the nth term, we can analyze the differences between consecutive terms.
Calculating Differences
- First, let's identify the differences between the terms:
- 7 - 1 = 6
- 25 - 7 = 18
- 79 - 25 = 54
- 241 - 79 = 162
- 727 - 241 = 486
- The differences are: 6, 18, 54, 162, 486.
- Now, let's calculate the second differences:
- 18 - 6 = 12
- 54 - 18 = 36
- 162 - 54 = 108
- 486 - 162 = 324
- The second differences are: 12, 36, 108, 324, which have a common ratio (3).
Identifying the Formula
The pattern in the differences suggests a geometric progression, hinting that the nth term has an exponential factor.
- Observing the terms, we can propose a formula of the form:
\[ a_n = 3^n - 2 \]
- Testing this formula with the values:
- For n=1: \( a_1 = 3^1 - 2 = 1 \)
- For n=2: \( a_2 = 3^2 - 2 = 7 \)
- For n=3: \( a_3 = 3^3 - 2 = 25 \)
- For n=4: \( a_4 = 3^4 - 2 = 79 \)
- For n=5: \( a_5 = 3^5 - 2 = 241 \)
- For n=6: \( a_6 = 3^6 - 2 = 727 \)
Conclusion
All terms match the original sequence, confirming that the correct answer is indeed:
**Option B: \( 3^n - 2 \)**
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For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {an} is ____________a)3n+1– 2b)3n– 2c)(-3)n+ 4d)n2– 2Correct answer is option 'B'. Can you explain this answer?
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