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The sequence of positive numbers $u_1, u_2, u_3, \ldots$, is given by $u_{n+1} = (u_n - 3)^2$, $u_1 = 1$ - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 1

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The-sequence-of-positive-numbers-$u_1,-u_2,-u_3,-\ldots$,-is-given-by---$u_{n+1}-=-(u_n---3)^2$,---$u_1-=-1$-Edexcel-A-Level Maths Pure-Question 4-2006-Paper 1.png

The sequence of positive numbers $u_1, u_2, u_3, \ldots$, is given by $u_{n+1} = (u_n - 3)^2$, $u_1 = 1$. (a) Find $u_2, u_3$ and $u_4$. (b) Write down the... show full transcript

Worked Solution & Example Answer:The sequence of positive numbers $u_1, u_2, u_3, \ldots$, is given by $u_{n+1} = (u_n - 3)^2$, $u_1 = 1$ - Edexcel - A-Level Maths Pure - Question 4 - 2006 - Paper 1

Step 1

Find $u_2$

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Answer

To find u2u_2, we substitute u1=1u_1 = 1 into the formula:

u2=(u1−3)2=(1−3)2=(−2)2=4.u_2 = (u_1 - 3)^2 = (1 - 3)^2 = (-2)^2 = 4.

Step 2

Find $u_3$

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Answer

Next, we find u3u_3 using u2u_2:

u3=(u2−3)2=(4−3)2=(1)2=1.u_3 = (u_2 - 3)^2 = (4 - 3)^2 = (1)^2 = 1.

Step 3

Find $u_4$

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Answer

Now, we calculate u4u_4:

u4=(u3−3)2=(1−3)2=(−2)2=4.u_4 = (u_3 - 3)^2 = (1 - 3)^2 = (-2)^2 = 4.

Step 4

Write down the value of $u_{10}$

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Answer

To find the pattern, notice that u2=4u_2 = 4, u3=1u_3 = 1, and u4=4u_4 = 4. The sequence oscillates between 4 and 1. Thus, since u10u_{10} is even,

u10=4.u_{10} = 4.

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