Given $u_1 = 1$, determine which one of the formulae below defines an increasing sequence for $n \geq 1$ - AQA - A-Level Maths Mechanics - Question 3 - 2019 - Paper 3
Question 3
Given $u_1 = 1$, determine which one of the formulae below defines an increasing sequence for $n \geq 1$.
Circle your answer.
$u_{n+1} = 1 + \frac{1}{u_n}$
$u_{n+... show full transcript
Worked Solution & Example Answer:Given $u_1 = 1$, determine which one of the formulae below defines an increasing sequence for $n \geq 1$ - AQA - A-Level Maths Mechanics - Question 3 - 2019 - Paper 3
Step 1
Determine if $u_{n+1} = 1 + \frac{1}{u_n}$ defines an increasing sequence
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Answer
This function will generate a sequence depending on un. However, substituting u1=1, gives u2=1+11=2, and so on, indicating an increasing sequence.
Step 2
Determine if $u_{n+1} = 2 - 0.9^{-1}$ defines an increasing sequence
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Answer
Calculating this expression, we find un+1=2−1.111..., which results in a constant value. Hence, not an increasing sequence.
Step 3
Determine if $u_{n+1} = -1 + 0.5u_n$ defines an increasing sequence
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Answer
Starting from u1=1, if we calculate u2=−1+0.5×1=−0.5. This sequence will not be increasing as it decreases.
Step 4
Determine if $u_n = 0.9^{-1}$ defines an increasing sequence
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Answer
This is a constant value of approximately 1.111..., which does not change and hence is not an increasing sequence.
