Question

in progresia aritmetica (an) n mai mare ca 1 , a10=3 , a3=7 .Calculati suma primilor zece

Answer 1

Formula sumei:
$S_n = \frac{(a_1 + a_n) * n}{2}$

Formula termenului general:
$a_k = a_1 + r * (k-1)$

a₁₀ = 3
=> a₁₀ = a₁ + 9r = 3
a₃ = 7
=> a₃ = a₁ + 2r = 7

a₁ + 9r = 3 => a₁ = 3 - 9r
a₁ + 2r = 7 => a₁ = 7 - 2r
=> 3 - 9r = 7 - 2r
=> 3 -7 = -2r + 9r
=> -4 = 7r
r = -4/7

$a_1 = 7 - 2r = 7 - 2\frac{-4}{7} = \frac{49 + 8}{7} = \frac{57}{7}$

a₁ = 57/7
a₁₀ = 3

Suma primilor 10 termeni:

$S_{10} = \frac{10 * (a_1 + a_{10})}{2}$

a₁ + a₁₀ = 57/7 + 3 = 46 / 7


$a_1 + a_{10} = \frac{57}{7} + 3 = \frac{57 + 21}{7} = \frac{78}{7}$

$S_{10} = \frac{10 * \frac{78}{7}}{2} = \frac{780}{7} : 2 = \\\\ = \frac{780}{7} * \frac{1}{2} = \frac{390}{7}$

Răspuns: 390/7