{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Les suites définies par récurrence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Attention : ne pas oublier d'exécuter la première cellule !**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Définition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u_0 = 3 et pour tout n>=0, u_n+1 = f(u_n).\n"
     ]
    }
   ],
   "source": [
    "nom=\"u\"\n",
    "nmin=0\n",
    "u_nmin=3\n",
    "\n",
    "def f(n) :\n",
    "    return 2*n+1\n",
    "\n",
    "print(f\"{nom}_{nmin} = {u_nmin} et pour tout n>={nmin}, {nom}_n+1 = f({nom}_n).\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calcul des premiers termes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u_ 0 =    3\n",
      "u_ 1 =    7\n",
      "u_ 2 =   15\n",
      "u_ 3 =   31\n",
      "u_ 4 =   63\n",
      "u_ 5 =  127\n",
      "u_ 6 =  255\n",
      "u_ 7 =  511\n",
      "u_ 8 = 1023\n",
      "u_ 9 = 2047\n",
      "u_10 = 4095\n"
     ]
    }
   ],
   "source": [
    "n=nmin\n",
    "u=u_nmin\n",
    "\n",
    "while(n<=10):\n",
    "    print(f\"{nom}_{n:2d} = {u:4d}\")\n",
    "    n=n+1\n",
    "    u=f(u)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calcul des premiers termes et des premières sommes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u_ 0 =    3   S_ 0 =    3\n",
      "u_ 1 =    7   S_ 1 =   10\n",
      "u_ 2 =   15   S_ 2 =   25\n",
      "u_ 3 =   31   S_ 3 =   56\n",
      "u_ 4 =   63   S_ 4 =  119\n",
      "u_ 5 =  127   S_ 5 =  246\n",
      "u_ 6 =  255   S_ 6 =  501\n",
      "u_ 7 =  511   S_ 7 = 1012\n",
      "u_ 8 = 1023   S_ 8 = 2035\n",
      "u_ 9 = 2047   S_ 9 = 4082\n",
      "u_10 = 4095   S_10 = 8177\n"
     ]
    }
   ],
   "source": [
    "n=nmin\n",
    "u=u_nmin\n",
    "s=u\n",
    "\n",
    "while(n<=10):\n",
    "    print(f\"{nom}_{n:2d} = {u:4d}   S_{n:2d} = {s:4d}\")\n",
    "    n=n+1\n",
    "    u=f(u)\n",
    "    s=s+u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test sur la valeur de la suite"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u_ 0 =    3\n",
      "u_ 1 =    7\n",
      "u_ 2 =   15\n",
      "u_ 3 =   31\n",
      "u_ 4 =   63\n",
      "\n",
      "------------------\n",
      "\n",
      "u_ 5 =  127\n"
     ]
    }
   ],
   "source": [
    "n=nmin\n",
    "u=u_nmin\n",
    "\n",
    "while(u<100):\n",
    "    print(f\"{nom}_{n:2d} = {u:4d}\")\n",
    "    n=n+1\n",
    "    u=f(u)\n",
    "\n",
    "print(\"\\n------------------\\n\")\n",
    "print(f\"{nom}_{n:2d} = {u:4d}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test sur la valeur de la somme"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u_ 0 =    3   S_ 0 =    3\n",
      "u_ 1 =    7   S_ 1 =   10\n",
      "u_ 2 =   15   S_ 2 =   25\n",
      "u_ 3 =   31   S_ 3 =   56\n",
      "u_ 4 =   63   S_ 4 =  119\n",
      "u_ 5 =  127   S_ 5 =  246\n",
      "u_ 6 =  255   S_ 6 =  501\n",
      "\n",
      "------------------\n",
      "\n",
      "u_ 7 =  511   S_ 7 = 1012\n"
     ]
    }
   ],
   "source": [
    "n=nmin\n",
    "u=u_nmin\n",
    "s=u\n",
    "\n",
    "while(s<1_000):\n",
    "    print(f\"{nom}_{n:2d} = {u:4d}   S_{n:2d} = {s:4d}\")\n",
    "    n=n+1\n",
    "    u=f(u)\n",
    "    s=s+u\n",
    "    \n",
    "print(\"\\n------------------\\n\")\n",
    "print(f\"{nom}_{n:2d} = {u:4d}   S_{n:2d} = {s:4d}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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