QuestionWrite the first four elements of the sequence. 1) $\frac{n+1}{3 n-1}$

Studdy Solution

STEP 1

What is this asking? We need to find the first four terms of a sequence defined by a formula involving n. Watch out! Don't forget that we start counting n from 1, not 0!
Also, be super careful with your arithmetic when plugging in values for n.

STEP 2

1. Plug in n = 1
2. Plug in n = 2
3. Plug in n = 3
4. Plug in n = 4

STEP 3

Let's kick things off by plugging in $n = 1$ into our formula $\frac{n+1}{3n-1}$ .
This gives us $\frac{1+1}{3 \cdot 1 - 1}$ .

STEP 4

Simplifying the numerator, $1 + 1 = 2$ .
The denominator is $3 \cdot 1 - 1 = 3 - 1 = 2$ .

STEP 5

So, the first term of our sequence is $\frac{2}{2} = 1$ .
Awesome!

STEP 6

Now, let's crank it up a notch and plug in $n = 2$ .
We get $\frac{2+1}{3 \cdot 2 - 1}$ .

STEP 7

The numerator is $2 + 1 = 3$ .
The denominator is $3 \cdot 2 - 1 = 6 - 1 = 5$ .

STEP 8

This gives us the second term: $\frac{3}{5}$ .
Fantastic!

STEP 9

Keep the momentum going with $n = 3$ !
We have $\frac{3+1}{3 \cdot 3 - 1}$ .

STEP 10

The numerator is $3 + 1 = 4$ .
The denominator is $3 \cdot 3 - 1 = 9 - 1 = 8$ .

STEP 11

So, the third term is $\frac{4}{8} = \frac{1}{2}$ .
Looking good!

STEP 12

Last one, let's plug in $n = 4$ .
We get $\frac{4+1}{3 \cdot 4 - 1}$ .

STEP 13

The numerator is $4 + 1 = 5$ .
The denominator is $3 \cdot 4 - 1 = 12 - 1 = 11$ .

STEP 14

This gives us our fourth term: $\frac{5}{11}$ .
We did it!

STEP 15

The first four terms of the sequence are $1$ , $\frac{3}{5}$ , $\frac{1}{2}$ , and $\frac{5}{11}$ .

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