Odpowiedź:
zad. 1
7+11+15+ ... +99
r = 2
a₁ = 7
aₙ = 99
[tex]S_{n} = \frac{a_{1} + a_{n}}{2} * n[/tex]
aₙ = a₁ + (n-1)r
99 = 7 + (n - 1) * 2
92 = (n -1) * 2
46 = n - 1
47 = n
[tex]S_{n} = \frac{7 + 99}{2} * 47 = \frac{106}{2} * 47 = 53 * 47 = 2491[/tex]
Odp. Suma = 2491
Zad. 2
a₂ = 6
a₇ = 192
S₈ = ?
a₇ = a₂ * q⁵
192 = 6 * q⁵
32 = q⁵
q = 2
a₁ = 3
aₙ = a₁ * qⁿ⁻¹
aₙ = 3 * 2ⁿ⁻¹
[tex]S_{8} = a_{1} * \frac{1 - q^{n}}{1-q}[/tex]
[tex]S_{8} = 3 * \frac{1 - 2^{8}}{1-2}[/tex]
[tex]S_{8} = 3 * \frac{1 - 256}{-1}[/tex]
[tex]S_{8} = 3 * \frac{ - 255}{-1}[/tex]
[tex]S_{8} = 3 * 255[/tex]
S⁸ = 765
Odp. ciąg: aₙ = 3 * 2ⁿ⁻¹, suma 8 wyrazów: S⁸ = 765.
zad. 3
q = 2
[tex]a_{7} = \frac{160}{3}[/tex]
a₁ = ?
aₙ = a₁ * qⁿ⁻¹
[tex]\frac{160}{3} = a_{1} * 2^{6}[/tex]
[tex]\frac{160}{3} = a_{1} * 64[/tex]
[tex]\frac{160}{3} * \frac{1}{64} = a_{1}[/tex]
[tex]a_{1} = \frac{160}{192} = \frac{5}{6}[/tex]
Odp. Pierwszy wyraz ciągu: [tex]a_{1} = \frac{160}{192} = \frac{5}{6}[/tex].
