A-Level Edexcel Maths Pure Trigonometric Equations: 4. (i) Show that $$\sum

4. (i) Show that $$\sum_{r=1}^{16} (3 + 5r + 2^r) = 131798$$ (ii) A sequence $u_n, u_2, u_3, \ldots$ is defined by $$u_{n+1} = \frac{1}{u_n}, \quad u_1 = \frac{2}{3}$$ Find the exact value of $$\sum_{r=1}^{100} u_r$$ - Edexcel - A-Level Maths Pure - Question 6 - 2018 - Paper 2

Question 6

$4.-(i)-Show-that-$$\sum_{r=1}^{16}-(3-+-5r-+-2^r)-=-131798$$--(ii)-A-sequence-$u_n,-u_2,-u_3,-\ldots$-is-defined-by-$$u_{n+1}-=-\frac{1}{u_n},-\quad-u_1-=-\frac{2}{3}$$--Find-the-exact-value-of-$$\sum_{r=1}^{100}-u_r$$-Edexcel-A-Level Maths Pure-Question 6-2018-Paper 2.png$

4. (i) Show that $$\sum_{r=1}^{16} (3 + 5r + 2^r) = 131798$$ (ii) A sequence $u_n, u_2, u_3, \ldots$ is defined by $$u_{n+1} = \frac{1}{u_n}, \quad u_1 = \frac{2}{3... show full transcript

Worked Solution & Example Answer:4. (i) Show that $$\sum_{r=1}^{16} (3 + 5r + 2^r) = 131798$$ (ii) A sequence $u_n, u_2, u_3, \ldots$ is defined by $$u_{n+1} = \frac{1}{u_n}, \quad u_1 = \frac{2}{3}$$ Find the exact value of $$\sum_{r=1}^{100} u_r$$ - Edexcel - A-Level Maths Pure - Question 6 - 2018 - Paper 2

Step 1

Show that $ \sum_{r=1}^{16} (3 + 5r + 2^r) = 131798 $

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Answer

To evaluate the sum, we can separate it into three distinct parts:

Constant Term Contribution: $\sum_{r=1}^{16} 3 = 3 \times 16 = 48$
Linear Term Contribution: $\sum_{r=1}^{16} 5r = 5 \sum_{r=1}^{16} r = 5 \times \frac{16(16+1)}{2} = 5 \times 136 = 680$
Exponential Term Contribution: $\sum_{r=1}^{16} 2^r = 2 (2^{16} - 1) \div (2-1) = 2(65536 - 1) = 131070$

Putting it all together:

$\sum_{r=1}^{16} (3 + 5r + 2^r) = 48 + 680 + 131070 = 131798$

Step 2

Find the exact value of $ \sum_{r=1}^{100} u_r $

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Answer

First, we derive the sequence values:

Compute Initial Terms:
- ( u_1 = \frac{2}{3} )
- ( u_2 = \frac{1}{u_1} = \frac{3}{2} )
- ( u_3 = \frac{1}{u_2} = \frac{2}{3} )
We notice that the sequence alternates:
- ( u_1 = \frac{2}{3}, u_2 = \frac{3}{2}, u_3 = \frac{2}{3}, u_4 = \frac{3}{2}, \ldots )
Pattern Identification:
- It can be established that for odd ( r ), ( u_r = \frac{2}{3} ) and for even ( r ), ( u_r = \frac{3}{2} ).
Sum Calculation:
- The total terms from 1 to 100 consist of 50 odd and 50 even terms:
- $\sum_{r=1}^{100} u_r = 50 \times \frac{2}{3} + 50 \times \frac{3}{2}$
- Simplifying further:
- $= \frac{100}{3} + \frac{150}{2} = \frac{100}{3} + 75 = \frac{100 + 225}{3} = \frac{325}{3}$

4. (i) Show that $$\sum_{r=1}^{16} (3 + 5r + 2^r) = 131798$$ (ii) A sequence $u_n, u_2, u_3, \ldots$ is defined by $$u_{n+1} = \frac{1}{u_n}, \quad u_1 = \frac{2}{3}$$ Find the exact value of $$\sum_{r=1}^{100} u_r$$ - Edexcel - A-Level Maths Pure - Question 6 - 2018 - Paper 2

Show that \( \sum_{r=1}^{16} (3 + 5r + 2^r) = 131798 \)

Find the exact value of \( \sum_{r=1}^{100} u_r \)

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