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- Calculate 3 first terms of the sequence
$(u_n)$ defined$\forall n \in \mathbb{N}$ by$u_n = 2n +1$
1, 3, 5
- Calculate first 4 terms of
$(u_n)$ $u_n = n^3$ 0, 1, 3, 8
- 10 first terms
0, 1, 3, 8, 27, 64, 125, 216, 343, 512
Let
- Calculate
$u_0$ and$u_1$ $u_0 = 5$ ,$u_1=4$
- 3 first terms,
$(u_n)$ , defined by$u_0=1$ and$u_{n+1}=2u_n+1$ 1, 3, 7
- 3 firsts terms,
$(v_n)$ , defined by$v_0=2$ and$v_{n+1}=v_n+n+3$ 2, 5, 9
- 3 first terms,
$(u_n)$ ,$u_0=2$ ,$u_{n+1}=u_n^2$ 2, 4, 16
Let
-
$w_1, w_2$ -1, 0
Sport club has 500 subs in 2019. Each year 80% stay subscribed and new 20 people subscribe.
- Subs in 2020
$$u_{n+1}=u_n\times\frac{4}{5}+20$$ $$u_{2O20}=500\times\frac{4}{5}+20$$ $$u_{2O20}=420$$ - Express
$u_{n+1}$ based on${u_n} $ $u_{n+1}=u_n\times\frac{4}{5}+20$$ - How many subs in 2030 (with calculator)
134
- When the gym hits 101 subs it will close, will it happen and when ?
Yes, in 2046
A company has 200 employees in 2019, each year 10% leave and 30 join.
- How many employees in 2020, 2021 ?
$$u_{n+1}=u_n\times\frac{9}{10}+30$$ $$u_{2020}=200\times\frac{9}{10}+30$$ $$u_{2020}=210$$ $$u_{2021}=210\times\frac{9}{10}+30$$ $$u_{2021}=219$$ - Express based on
$n$ $$u_{n+1}=u_n\times\frac{9}{10}+30$$
In a shop a sweater costs 60$, each period of sale the price decreases by 15%.
-
Price after first period of sale
$$u_{n+1}=u_n\times \frac{85}{100}$$ $$u_1=60\times \frac{85}{100}$$ $$u_1=51$$ -
We note
$(u_n)$ the price of the sweater after$n$ period of sale.-
a.
$u_0$ and$u_1$ 60, 51
-
b. Express based on
$n$ $$u_{n+1}=u_n\times \frac{85}{100}$$
-
Represent graphically, the 5 first terms of the sequence
-
Let,
$(u_n)$ an artihmetic sequence of first term$u_0=5$ and of common difference$-2$ -
a. Give the expression of
$u_n$ $u_{n+1}=u_n-2n$ -
b. Give value of
$u_{25}$ $u_{25}=5-2\times{25}=-45$
-
-
Are the following sequences arithmetic
$$v_n=3n-2$$ yes
$v_0=-2$ $v_1=1$ $v_2=4$ $v_1-v_0 = 3 = v_2-v_1$ Demonstration
$$v_{n+1}-v_n=[3(n+1)-2]-(3n-2)$$ >$$=3n+1-3n+2$$ >$$v_{n+1}-v_n=3$$ $$w_n=n^2+1$$ no, it is geometric$$v_{n+1}-v_n=[(n+1)^2+1]-(n^2+1)$$ >$$=n^2+2n+1-n^2-1$$ >$$v_{n+1}-v_n=2n$$ $$w_n=\frac{n^2+n}{n}$$ $$w_n=\frac{n^2+n}{n} = \frac{\frac{1}{n}(n^2+n)}{1} = n + 1$$ $$w_{n+1}-w_n=(n+2)-(n+1)$$ $$w_{n+1}-w_n=1$$
Let,
- Give the expression of
$u_n$ $u_{n+1}=u_n+3$ - Indentify the value of
$u_{20}$ $u_{20}=-2+20\times{3}=58$
Are the following suites arithmetic
-
$v_n=\sqrt{n}$ no
$$\sqrt{2} - \sqrt{1} \ne \sqrt{3} - \sqrt{2}$$ -
$w_n=-n+4$ yes
$$w_{n+1}-w_n=(-n-1+4)-(-n+4)$$ >$$w_{n+1}-w_n=-n+3+n-4$$ >$$w_{n+1}-w_n=-1$$
Let
- Give the expression
$u_n$ depending on$n$
- a.
$u_n$ $$\forall n \in \mathbb{N}, \forall p \in \mathbb{N} \space u_{n+p}=u_p\times{q^{n-p}}$$ $$u_n=\frac{1}{4}\times{2^{n-2}}$$ - b. Value of
$u_6$ $$u_6=\frac{1}{4}\times{2^4} = 4$$
- Are the following sequences geometric
$$u_n=n^2+1$$ No
$$u_0=0+1=1$$ >$$u_1=1+1=2$$ >$$u_2=4+1=5$$ >$$\frac{u_0}{u_1}\ne\frac{u_1}{u_2}$$ $$u_n=2^{n+1}$$ yes$$\frac{u_{n+1}}{u_n}=\frac{2^{n+2}}{2^{n+1}}=2$$ $$u_n=\frac{1}{n}$$ No$$u_1=\frac{1}{1}=1$$ >$$u_2=\frac{1}{2}$$ >$$u_3=\frac{1}{3}$$ >$$\frac{u_2}{u_1}\ne\frac{u_3}{u_2}$$
Let
- Give the expression of
$u_n$ based on$n$ $$u_n=-2\times{2^n}$$ - What is the value of
$u_{10}$ $$u_{10}=-2\times{2^{10}}=–2\times{1024}=-2048$$
Are the following sequences geometric?
No
$$u_0=0$$ >$$u_1=1$$ >$$u_2=\sqrt{2}$$ >$$\frac{u_1}{u_0}\ne\frac{u_2}{u_1}$$ $$w_n=\frac{1}{3^n}$$ Yes$$\frac{w_{n+1}}{w_n}=\frac{\frac{1}{3^{n+1}}}{\frac{1}{3^n}}=\frac{\frac{1}{3}\times{3^{n+1}}}{\frac{1}{3}\times{3^n}}=3$$
- Calculate the sum of
$101+102+103+...+998+999$ $$1+2+...+n=\frac{n(n+1)}{2}$$ $$\frac{999(999+1)}{2} - \frac{100(100+1)}{2} = 494,450$$ -
$(u_n)$ is an arithmetic sequence defined by$u_0=2$ and of common difference 3. Calculate the sum$S=u_0+u_1+...+u_{15}$ $$S=15 \times \frac{u_0+u_{15}}{2}$$ $$S=15 \times \frac{2+47}{2}$$ $$S=392$$ 
-
$(u_n)$ a geometric sequence defined by$u_0=3$ and of common difference 2. Calculate the sum of the first 20 terms.$$S=u_0+u_1+...+u_{19}$$ $$u_n=u_0\times{q^n}$$ $$S = 3 + (3\times{2^1}) + ... + (3\times{2^{19}})$$ $$S = 3(2^0+2^1+...+2^{19})$$ $$S = 3 \times \frac{1-q^{n+1}}{1-q}$$ $$S = 3 \times \frac{1-2^{20}}{-1}$$ $$S = 3,145,725$$
Calculate the following sums
$1+2+...+150 = \large\frac{150\times 151}{2}\small = 11,325$ $50 + 51 + ... + 150 = \large\frac{150\times 151}{2} \small- \large\frac{49\times 50}{2}\small = 10,100$
Calculate the following sums
- a)
$1 + \frac{1}{2} + \frac{1}{2^2} + ... + \frac{1}{2^{16}}$ $$\frac{\frac{1}{2^{16}}(\frac{1}{2^{16}}+1)}{2} = 7\times10^{-6}$$ - b)
$1 + \frac{1}{2} + \frac{1}{2^2} + ... + \frac{1}{2^{10}}$ $$\frac{\frac{1}{2^{10}}(\frac{1}{2^{10}}+1)}{2} = 488\times10^{-6}$$ - b)
$\frac{1}{2^{11}} + \frac{1}{2^{12}} + ... + \frac{1}{2^{16}}$ $$-4.81\times 10^{-4}$$
-
Let
$(u_n)$ be an arithmetic sequence defined by$u_0=-2$ and$d = 4$ . Calculate the sum of the first 21 terms.$$u_n = u_0 + d\times n$$ $$S = u_0 + (u_0+d\times 1) + (u_0+d\times 2) + ... + (u_0+d\times 20)$$ $$S = 21 \times u_0 + (d+2d+...+20d)$$ $$S = 21 \times u_0 + d \times (1+2+...+20)$$ $$S = 21 \times -2 + 4 \times \frac{20 \times 21}{2}$$ $$S = -38 \times 210$$ $$S = -7,980$$ -
Let
$(u_n)$ be a geometrical sequence defined by$u_0=\frac{1}{2}$ and common ratio 3. Calculate the sum of the first 10 terms.
- Calculate
$u_0+u_1+...+u_{50}$ $$S = 51\times\frac{-142}{2} = -3,621$$ - Sum, 20 first terms
$$S = 20\times\frac{-49}{2} = -490$$ - Guess the sum of
$u_{20}+...u_{50}$ $$-3,621 + 490 = -3131$$
Study the variations of the sequence
a)