function series_recursive($level)
{
if ($level <= 3) {
$a1 = pow(-1, rand(1, 2)) * rand(1, 4);
// initial value of a1
$a2 = 0;
// initial value of a2
$coeff_an_1 = pow(-1, rand(1, 2)) * rand(1, 3);
// coefficient of a_{n-1} in recursive formula
$coeff_an_2 = 0;
// coefficient of a_{n-1} in recursive formula
$coeff_n = 0;
// coefficient of n in recursive formula
$pow_n = 0;
// power of n in recursive formula
$coeff_a0 = pow(-1, rand(1, 2)) * rand(1, 5);
// constant term in recursive formula
$index = 2;
// index of member we are looking for
$formula = ' ' . $coeff_an_1 . '\\cdot a_{n-1}' . '+ ' . $coeff_a0;
// recursive formula
$init = '$a_1=' . $a1 . '$';
// initial values
} elseif ($level <= 6) {
$a1 = pow(-1, rand(1, 2)) * rand(5, 10);
$a2 = 0;
$coeff_an_1 = pow(-1, rand(1, 2)) * rand(4, 7);
$coeff_an_2 = 0;
$coeff_n = pow(-1, rand(1, 2)) * rand(0, 2);
$pow_n = rand(1, 3);
$coeff_a0 = pow(-1, rand(1, 2)) * rand(5, 10);
$index = rand(2, 3);
$formula = ' ' . $coeff_an_1 . '\\cdot a_{n-1}' . '+ ' . $coeff_n . '\\cdot n^' . $pow_n . '+ ' . $coeff_a0;
$init = '$a_1=' . $a1 . '$';
} else {
$a1 = pow(-1, rand(1, 2)) * rand(2, 4);
$a2 = pow(-1, rand(1, 2)) * rand(1, 3);
$coeff_an_1 = pow(-1, rand(1, 2)) * rand(4, 7);
$coeff_an_2 = pow(-1, rand(1, 2)) * rand(2, 5);
$coeff_n = pow(-1, rand(1, 2)) * rand(2, 5);
$pow_n = rand(1, 5);
$coeff_a0 = pow(-1, rand(1, 2)) * rand(5, 10);
$index = rand(3, 5);
$formula = ' ' . $coeff_an_1 . '\\cdot a_{n-1}' . '+ ' . $coeff_an_2 . '\\cdot a_{n-2}' . '+ ' . $coeff_n . '\\cdot n^' . $pow_n . '+ ' . $coeff_a0;
$init = '$a_1=' . $a1 . '$ és $a_2=' . $a2 . '$';
}
$formula = str_replace(' 1\\cdot ', '', $formula);
$formula = str_replace(' -1\\cdot ', ' -', $formula);
$formula = str_replace('+ -', '-', $formula);
$formula = str_replace('^1+', '+', $formula);
$formula2 = str_replace('\\', '\\\\', $formula);
$correct = recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index);
if ($correct > 9999) {
$correct = number_format($correct, 0, ',', '\\,');
}
$an_1 = recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index - 1);
if ($index > 2 && $coeff_an_2 != 0) {
$an_2 = recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index - 2);
if ($an_2 > 0) {
$formula2 = str_replace('a_{n-2}', $an_2, $formula2);
} else {
$formula2 = str_replace('a_{n-2}', '(' . $an_2 . ')', $formula2);
}
}
if ($an_1 > 0) {
$formula2 = str_replace('a_{n-1}', $an_1, $formula2);
} else {
$formula2 = str_replace('a_{n-1}', '(' . $an_1 . ')', $formula2);
}
$formula2 = str_replace('n', $index, $formula2);
$solution = '$' . $correct . '$ (mert $' . $formula2 . '=' . $correct . '$)';
$question = 'Egy sorozat képzési szabálya a következő:$$a_n=' . $formula . '$$' . 'Ha tudjuk, hogy ' . $init . ', akkor mekkora $a_{' . $index . '}$ értéke?';
return array('question' => $question, 'correct' => $correct, 'solution' => $solution);
}
/**
* Get member of series defined by recursive formula
*
* Formula: $coeff_an_1*a_{n-1} + $coeff_an_2*a_{n-2} + $coeff_n*n^{pow_n} + $coeff_a0
*
* @param int $a1 Initial value of a1
* @param int $a2 Initial value of a2
* @param int $coeff_an_1 Coefficient of a_{n-1} in recursive formula
* @param int $coeff_an_2 Coefficient of a_{n-2} in recursive formula
* @param int $coeff_n Coefficient of n in recursive formula
* @param int $pow_n Power of n in recursive formula
* @param int $coeff_a0 Constant term in recursive formula
* @param int $index Index of member we are looking for
*
* @return int $res Result
*/
function recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index)
{
if ($index == 1) {
$res = $a1;
} elseif ($index == 2 && $coeff_an_2 != 0) {
$res = $a2;
} else {
$an_1 = recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index - 1);
if ($coeff_an_2 != 0) {
$an_2 = recursiveSeries($a1, $a2, $coeff_an_1, $coeff_an_2, $coeff_n, $pow_n, $coeff_a0, $index - 2);
} else {
$an_2 = 0;
}
$res = $coeff_an_1 * $an_1 + $coeff_an_2 * $an_2 + $coeff_n * pow($index, $pow_n) + $coeff_a0;
}
return $res;
}