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(a) Calculate the 10th term.
$$F_{10} = \text{.................................................................}$$
(b) (i) Complete the table.
| $$F_2$$ | $$= 3$$ | $$F_3 - F_1 = 3$$ |
|-----------|---------------|-----------------|
| $$F_2 + F_4 = \text{..........}$$ | $$F_5 - F_1 = 14$$ |
| $$F_2 + F_4 + F_6 = \text{..........}$$ | $$F_7 - F_1 = \text{..........}$$ |
| $$F_2 + F_4 + F_6 + F_8 = 123$$ | $$F_9 - F_1 = \text{..........}$$ |
(b) (ii) Complete this statement.
$$F_2 + F_4 + F_6 + F_8 + F_{10} = F - F \text{........}$$
(c) (i) Complete the table.
| $$F_1$$ | $$= 5$$ | $$F_2 + F_1 - F_2 = 5$$ |
|------------|-----------------|-----------------|
| $$F_1 + F_3 = \text{..........}$$ | $$F_4 + F_1 - F_2 = 13$$ |
| $$F_1 + F_3 + F_5 = \text{..........}$$ | $$F_6 + F_1 - F_2 = \text{..........}$$ |
| $$F_1 + F_3 + F_5 + F_7 = 81$$ | $$F_8 + F_1 - F_2 = \text{..........}$$ |
(c) (ii) Complete this statement.
$$F_1 + F_3 + F_5 + F_7 + F_9 = F + F \text{........} - F \text{........}$$
(d) Use your statements in part (b)(ii) and part (c)(ii), and the definition of an F-type sequence, to show that
$$F_1 + F_2 + F_3 + F_4 + F_5 + F_6 + F_7 + F_8 + F_9 + F_{10} = F_{12} - F_2$$
(e) Use the statement in part (d) to complete this general statement.
$$F_1 + F_2 + F_3 + \cdots + F_n = F \text{........} - F \text{........}$$
558
517
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In another F-type sequence the first term is 3 and the second term is 1.
(a) Complete the first five terms.
3, 1, ......., ......., ....... [1]
(b) Is your statement in Question 1(e) correct for the sum of the first five terms in this sequence?
........................................................................................................ [3]
543
582
529
In another F-type sequence the 2nd term is 3 and the 12th term is 652.
(a) Use your answer to Question 1(e) to find the sum of the first 10 terms.
.................................................. [2]
(b) The sum of the first 12 terms of this sequence is 1704.
Find the 10th term.
.................................................. [3]
502
530
532
The Fibonacci sequence is a special F-type sequence.
The sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots
(a) Use this information and your answers to Question 1(c) to simplify this sum.
\( F_1 + F_3 + F_5 + \cdots + F_{2n-1} \)
..................................................... [1]
(b) The 16th term in the Fibonacci sequence is 987.
Find the 8 different terms in the Fibonacci sequence that add up to 987.
......................................................................................................................... [2]
512
555
571
The first four terms of an F-type sequence are $a$, $b$, $c$ and $d$.
(a) There is a relationship between $c^2 - b^2$ and a simple combination of $a$ and $d$.
Investigate this relationship by making up at least three numerical examples of F-type sequences.
Write down this relationship.
................................................................................................................................................................................. [4]
(b) The first term of the F-type sequence is $a$ and the second term is $b$.
(i) Write $c$ and $d$ in terms of $a$ and $b$, in their simplest form.
$c = ........................................................... $
$d = ........................................................... $ [1]
(ii) Use algebra to show that the relationship in part (a) is correct. [2]
568
535
581
This task looks at the age, $a$, of a goat and its biological age, $b$, when compared to a human.
A goat's body ages more quickly than a human body.
At birth, a goat's age and its biological age are both 0.
When $a = 0$ then $b = 0$.
The life expectancy for a human is 73.5 years.
The life expectancy for a goat is 10.5 years, which matches the biological life expectancy of 73.5 years for a human.
When $a = 10.5$ then $b = 73.5$.
(a) Find a straight-line model, in its simplest form, for $b$ in terms of $a$.
This is Model M.
.................................................. [3]
(b) Sketch the graph of your model.
[2]
(c) A goat is 8 years old, so $a = 8$.
Find its biological age, $b$.
.................................................. [1]
531
534
572
Goats age more quickly when young.
A goat that is 2 years old has a biological age of 24 years.
So, when $a = 2, b = 24$.
(a) Find a straight-line model for $b$ in terms of $a$ for $0 \leq a \leq 2$.
.......................................................... [1]
(b) After a goat reaches the age of 2 years, its biological age increases by 4 each year.
(i) Find its biological age, $b$, when $a = 10$.
.......................................................... [2]
(ii) Find a straight-line model for $b$ in terms of $a$ for $a \geq 2$.
Write the model in its simplest form.
This is extbf{Model N}.
.......................................................... [3]
(c) Sketch the graphs of your straight-line models in extit{part (a)} and extit{part (b)(ii)} on the axes on page 8.
.......................................................... [2]
562
552
587
(a) Use the graph to write down the biological age of a goat that is:
• 2 years old .....................................................
• 10 years old. .................................................. [1]
(b) This model for the biological age is $b = g \log a + h$ where $g$ and $h$ are constants.
(i) Use your answers to part (a) to write down two equations in $g$ and $h$.
.......................................................
....................................................... [1]
(ii) Use algebra to find $g$ and $h$, correct to the nearest integer. Write down the model.
This is Model P.
....................................................... [3]
(c) Find the age, correct to one decimal place, of a goat whose biological age is 70.
.......................................................... [3]
576
500
592
A goat lives until it is 18 years old, which is old for a goat.
For each model calculate the biological age of the goat.
Write down whether each model is valid or not valid for this goat.
Model M in Question 6(a)
...............................................................
Model N in Question 7(b)(ii)
...............................................................
Model P in Question 8(b)(ii)
...............................................................
537
537
598
Find the ages between which
biological age from Model N \textless{} biological age from Model P \textless{} biological age from Model M.
Between \text{.....................} and \text{.....................}
589
599
541
