Find the Number of Terms in the Sequence 2n-1: Odd Number Series

Arithmetic Sequences with Pattern Recognition

The series 2n1 2n-1 contains ___ numbers.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Complete
00:05 Let's find the first element
00:08 Now let's find the second element
00:13 And the third element
00:20 Let's identify the sequence pattern and find the remaining elements
00:28 We can see that all numbers are odd
00:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

The series 2n1 2n-1 contains ___ numbers.

2

Step-by-step solution

The sequence given by 2n1 2n-1 is a series of numbers.

Let's evaluate this sequence by calculating the first few terms:

  • When n=1 n = 1 , 2(1)1=1 2(1) - 1 = 1
  • When n=2 n = 2 , 2(2)1=3 2(2) - 1 = 3
  • When n=3 n = 3 , 2(3)1=5 2(3) - 1 = 5
  • When n=4 n = 4 , 2(4)1=7 2(4) - 1 = 7

From these calculations, the sequence produced is 1,3,5,7, 1, 3, 5, 7, \ldots , which are all odd numbers.

Therefore, the numbers contained in the series 2n1 2n-1 are Odd numbers.

The correct answer, according to the provided choices, is: Odd.

3

Final Answer

Odd

Key Points to Remember

Essential concepts to master this topic
  • Formula Pattern: The expression 2n1 2n-1 generates consecutive odd numbers
  • Substitution Method: Substitute n = 1, 2, 3 to get 1, 3, 5
  • Verify Pattern: Check that all results are odd by confirming each is not divisible by 2 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing the formula with the pattern it generates
    Don't think 2n1 2n-1 itself is the answer = missing the pattern! The formula is just instructions. Always substitute values for n to see what type of numbers the sequence actually produces.

Practice Quiz

Test your knowledge with interactive questions

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

FAQ

Everything you need to know about this question

Why does 2n1 2n-1 always give odd numbers?

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Mathematical reason: When you multiply any integer by 2, you get an even number. Subtracting 1 from an even number always gives an odd number!

How do I know what type of numbers a sequence produces?

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Substitute the first few values: n = 1, 2, 3, 4. Look at the results and identify the pattern. Are they all even? All odd? All positive?

What if n can be any positive integer?

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The sequence continues infinitely! 2n1 2n-1 will generate all positive odd numbers: 1, 3, 5, 7, 9, 11, 13...

Could this sequence contain negative numbers?

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Only if n can be zero or negative. For positive integers n, 2n1 2n-1 gives positive odd numbers starting from 1.

Is there a faster way than substituting each value?

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Yes! Recognize that 2n is always even, so 2n - 1 is always odd. This algebraic reasoning shows the pattern without calculations.

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