Math  /  Algebra

QuestionAn iterative formula is shown below. xn+1=8xn5x_{n+1}=\sqrt{8 x_{n}-5}
Starting with x1=3x_{1}=3, calculate the values of x2,x3x_{2}, x_{3} and x4x_{4}. Give your answers to 3 d.p.

Studdy Solution

STEP 1

What is this asking? We're starting with a number, plugging it into a formula, and then using the result to plug back into the formula a few more times.
We want to find the next three numbers! Watch out! Don't forget to round to **three decimal places** after each calculation!

STEP 2

1. Start with the initial value
2. Calculate the second value
3. Calculate the third value
4. Calculate the fourth value

STEP 3

Alright, let's kick things off with our **initial value**.
We're given x1=3 x_1 = 3 .
This is the number we'll use to start our calculations.
It's like the first domino in a chain reaction!

STEP 4

Now, let's **calculate the second value** using the formula xn+1=8xn5 x_{n+1} = \sqrt{8x_n - 5} .
Plug in x1=3 x_1 = 3 :
x2=835x_2 = \sqrt{8 \cdot 3 - 5}

STEP 5

Simplify inside the square root:
x2=245=19x_2 = \sqrt{24 - 5} = \sqrt{19}

STEP 6

Now, find the square root of 19 and round it to three decimal places:
x24.359x_2 \approx 4.359

STEP 7

Let's move on to **calculate the third value**.
Use x24.359 x_2 \approx 4.359 in the formula:
x3=84.3595x_3 = \sqrt{8 \cdot 4.359 - 5}

STEP 8

Simplify inside the square root:
x3=34.8725=29.872x_3 = \sqrt{34.872 - 5} = \sqrt{29.872}

STEP 9

Find the square root of 29.872 and round it to three decimal places:
x35.466x_3 \approx 5.466

STEP 10

Finally, let's **calculate the fourth value**.
Use x35.466 x_3 \approx 5.466 in the formula:
x4=85.4665x_4 = \sqrt{8 \cdot 5.466 - 5}

STEP 11

Simplify inside the square root:
x4=43.7285=38.728x_4 = \sqrt{43.728 - 5} = \sqrt{38.728}

STEP 12

Find the square root of 38.728 and round it to three decimal places:
x46.224x_4 \approx 6.224

STEP 13

The values are: - x24.359 x_2 \approx 4.359 - x35.466 x_3 \approx 5.466 - x46.224 x_4 \approx 6.224

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord