Numerical sequences: corrected high school math exercises in PDF.

/ Math exercises in the final year of high school / By Signaler une erreur

A series of math exercises for the final year of high school on numerical sequences.

This sheet involves the following concepts:

definition of a sequence;
sum of the terms of a sequence;
convergence of a numerical sequence;
asymptotic behavior of a sequence;
study suite and functions;
recurring suites.

Exercise #1:

u is the sequence defined by and, for any natural number n, $u_{n+1}=\sqrt{u_n^2+1}$ .

With the spreadsheet, the first values of and were obtained below.

Conjecture an expression for as a function of n.
Validate this conjecture by reasoning by recurrence.

Exercise #2:

V is the sequence defined by and for any natural number n, $V_{n+1}=V_n+2n+2$ .

Prove by recurrence that for any natural number n, .

Exercise #3:

Show by recurrence that, for any natural number n, $2^n\geq\,\,\,\,n+1$ .

Exercise #4:

In this figure:

the triangles are rectangles.

Prove by recurrence that for any natural number n, $OA_n=\sqrt{4n+1}$ .

Exercise #5:

Study, justifying, the limit in infinity of each of the following numerical series:

$1.\,u_n=3(2-0,9^n)\\2.\,v_n=1,01^n-5\\3.\,w_n=\frac{3+0,2^n}{0,9^n-5}\\4.\,t_n=\frac{4^n+5}{2\times \,3^n}$

Exercise #6:

u is the geometric sequence of reason 0.8 and first term .

For any non-zero natural number n, express as a function of n.
Study the limit of the sequence $\,(\,S_n\,\,)$ .

Exercise #7:

Consider the sequence defined by and for all $n\in\,\mathbb{N}$ ,

$u_{n+1}=\frac{3u_n}{1+2u_n}$ .

1)Let f be the function defined on $%5B0;+\infty%5B$ by $f(x)=\frac{3x}{1+2x}$ .

a) Study the variations of f on $%5B0;+\infty%5B$ .

b) Deduce that if $x\in%5B0;1%5D$ , then f ‘ (x) $\in%5B0;1%5D$ .

2)Prove by recurrence that, for any natural number n, $0\leq\,\,u_n\leq\,\,1$ .

3)Determine the direction of variation of the sequence .

Exercise #8:

The sequence is defined by and for all $n\in\,\mathbb{N}^*$ ,

$u_{n+1}=u_n+2n+1$ .

1)Using a calculator or spreadsheet, determine the first ten

terms of the sequence .

2)a)What conjecture can be made about the expression of as a function of n?

b) Prove this conjecture by recurrence.

Exercise #9:

Show by recurrence that, for any natural number n not zero,

$\sum_{q=1}^{n}q^2=\frac{n(n+1)(2n+1)}{6}$ .

Exercise #10:

Determine the limit of defined on $\mathbb{N}^*$ using general theorems.

.

$2)u_n=\frac{3}{3+2n}$ .

$3)u_n=4n-1+\frac{5}{\sqrt{n}}$ .

$4)u_n=-n^2-5n+\frac{1}{n}$

Exercise #11:

Let be the sequence defined by and, for all $n\in\,\mathbb{N}$ ,

$u_{n+1}=-\frac{1}{3}u_n+1$ .

Let be the sequence defined for any natural number n by :

.

1) Show that the sequence is geometric of reason $-\frac{1}{3}$ .

Specify the first term.

2) Determine the expression of as a function of n and deduce that,

for any natural number n :

$u_n=\frac{17}{4}\times \,(-\frac{1}{3})^n+\frac{3}{4}$ .

3) Determine the limit of the sequence .

Exercise #12:

Investigate whether the following sequences, defined on $\mathbb{N}$ , are bounded.

$1)u_n=(\frac{1}{3})^n-8$ .

.

.

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