Find the Term-to-Term Rule: Analyzing the Sequence 2, 2½, 3⅓, 4¼

Question

Look at the following sequence:

2,212,313,414 2,2\frac{1}{2},3\frac{1}{3},4\frac{1}{4}\ldots

Which expression represents the term-to-term rule of the sequence?

Video Solution

Step-by-Step Solution

To find the expression that accurately represents the sequence, let's analyze the given numbers:

  • The first term is 2 2 , which can be represented as 2+11=2+11 2 + \frac{1}{1} = 2 + \frac{1}{1} for n=2 n = 2 .
  • The second term is 212=2+12 2\frac{1}{2} = 2 + \frac{1}{2} , when n=2 n = 2 .
  • The third term is 313=3+13 3\frac{1}{3} = 3 + \frac{1}{3} , when n=3 n = 3 .
  • The fourth term is 414=4+14 4\frac{1}{4} = 4 + \frac{1}{4} , when n=4 n = 4 .

From the sequence pattern, we see that each term is indeed n+1n n + \frac{1}{n} .

Now, let's express each choice based on n n :

  • Choice 1: 2n+1n 2n + \frac{1}{n} - This does not match our pattern.
  • Choice 2: 10.5n+2n+10.5nn \frac{1}{0.5n} + 2n + \frac{1}{0.5n} - n - This form is complex and incorrect.
  • Choice 3: Indicates there is no correct property, but we identified one.
  • Choice 4: n2+2n+3nn21n2n n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n simplifies to n+1n n + \frac{1}{n} .

Therefore, the expression n2+2n+3nn21n2n n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n simplifies correctly to describe the term-to-term rule of the sequence.

The solution to the problem is the expression: n2+2n+3nn21n2n n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n .

Answer

n2+2n+3nn21n2n n^2+\frac{2}{n}+3n-n^2-\frac{1}{n}-2n