Here is a formula for an iterative process.
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uₙ₊₁ = 24/uₙ + 4
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u₂ = 8
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Work out the values of u₁ and u₃.
How to approach this question
1. **To find u₃:** This is a "forward" calculation. Use the given formula uₙ₊₁ = 24/uₙ + 4. Let n=2. This gives you the formula for u₃ in terms of u₂. Substitute the given value of u₂ = 8 into this formula and calculate the result.
2. **To find u₁:** This is a "backward" calculation. Use the formula again, but this time let n=1. This gives you u₂ = 24/u₁ + 4. You know the value of u₂, so you can substitute it in. This leaves you with an equation with u₁ as the unknown. Rearrange and solve this equation to find u₁.
Full Answer
**Finding u₃ (forwards):**
To find u₃, we use the formula with n=2. This means we substitute u₂ into the formula.
u₃ = 24/u₂ + 4
We are given u₂ = 8.
u₃ = 24/8 + 4
u₃ = 3 + 4
**u₃ = 7**
**Finding u₁ (backwards):**
To find u₁, we use the formula with n=1.
u₂ = 24/u₁ + 4
We know u₂ = 8, so we can set up an equation to solve for u₁.
8 = 24/u₁ + 4
Subtract 4 from both sides:
4 = 24/u₁
Multiply both sides by u₁:
4u₁ = 24
Divide by 4:
u₁ = 24 / 4
**u₁ = 6**
**Answers: u₁ = 6 and u₃ = 7**
The iterative formula is uₙ₊₁ = 24/uₙ + 4. We are given u₂ = 8.
**To find u₃:**
We need to find the term after u₂. So we set n=2 in the formula:
u₂₊₁ = u₃ = 24/u₂ + 4
Substitute the value of u₂ = 8:
u₃ = 24/8 + 4
u₃ = 3 + 4
u₃ = 7
**To find u₁:**
We need to find the term before u₂. So we set n=1 in the formula:
u₁₊₁ = u₂ = 24/u₁ + 4
We know u₂ = 8, so we can write an equation:
8 = 24/u₁ + 4
Now, we solve this equation for u₁.
Subtract 4 from both sides:
8 - 4 = 24/u₁
4 = 24/u₁
Multiply both sides by u₁:
4 * u₁ = 24
Divide both sides by 4:
u₁ = 24 / 4
u₁ = 6
So, the values are u₁ = 6 and u₃ = 7.
Common mistakes
✗ Forgetting the order of operations (BIDMAS/BODMAS), e.g., adding 4 before dividing.
✗ Errors in rearranging the equation to find u₁. For example, incorrectly isolating u₁.
✗ Confusing the subscripts and substituting the wrong values.
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