Consecutive terms of a sequence are related by
$$u_{n+1} = 3 - (u_n)^2$$
In the case that $u_1 = 2$:
7 (a) (i) Find $u_3$
7 (a) (ii) Find $u_{50}$
7 (b) State a different value for $u_1$ which gives the same value for $u_{50}$ as found in part (a)(ii). - AQA - A-Level Maths Mechanics - Question 7 - 2020 - Paper 1
Question 7
Consecutive terms of a sequence are related by
$$u_{n+1} = 3 - (u_n)^2$$
In the case that $u_1 = 2$:
7 (a) (i) Find $u_3$
7 (a) (ii) Find $u_{50}$
7 (b) State a... show full transcript
Worked Solution & Example Answer:Consecutive terms of a sequence are related by
$$u_{n+1} = 3 - (u_n)^2$$
In the case that $u_1 = 2$:
7 (a) (i) Find $u_3$
7 (a) (ii) Find $u_{50}$
7 (b) State a different value for $u_1$ which gives the same value for $u_{50}$ as found in part (a)(ii). - AQA - A-Level Maths Mechanics - Question 7 - 2020 - Paper 1
Step 1
Find $u_3$
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Answer
To find u3, we first need to compute u2 using the sequence formula.
Given that u1=2:
u2=3−(u1)2=3−(2)2=3−4=−1
Now that we have u2, we can calculate u3:
u3=3−(u2)2=3−(−1)2=3−1=2
Thus, u3=2.
Step 2
Find $u_{50}$
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Answer
To find u50, we notice that we obtained u2=−1 and u3=2. The sequence appears to alternate between -1 and 2:
u2=−1
u3=2
u4=3−(u3)2=3−4=−1
u5=3−(u4)2=3−1=2
Therefore, it follows that:
For even n, un=−1
For odd n, un=2
Since 50 is even:
u50=−1
Step 3
State a different value for $u_1$ which gives the same value for $u_{50}$ as found in part (a)(ii)
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Answer
To achieve the same outcome for u50, we can use u_1 = - rac{ ext{sqrt{2}}}{2}. This values can lead back to u50=−1 as illustrated during the calculation of the sequence. Therefore, any values that satisfy the equation could also work, such as u1=2 or u1=−2, as both will yield the same result for u50.
