Odpowiedzi:
11. a)
x + (x + 2) + ( x + 4) + ... + ( x + 24) = 142
Ciąg arytmetyczny:
[tex]a_1 = x[/tex] r = 2 [tex]a_n = x + 24[/tex]
[tex]S_n = 142[/tex]
więc
[tex]a_n = a_1 + ( n - 1)*r = x + ( n - 1)*2[/tex]
x + ( n - 1)*2 = x + 24
( n - 1)*2 = 24
n - 1 = 12
n = 13
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[tex]S_{13} = \frac{a_1 + a_{13}}{2} * 13 = 142[/tex]
[tex]\frac{x + x + 24}{2} *13 = 142[/tex]
( x + 12)*13 = 142
13 x + 156 = 142
13 x = - 14
x = - [tex]\frac{14}{13}[/tex]
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b) 7 + 9 + 11 + ... + x = 432
Ciąg arytmetyczny: [tex]a_1 = 7[/tex] r = 2 [tex]a_n = x\\S_n = 432[/tex]
więc
x = [tex]a_n =a_1 + ( n -1 )*r = 7 + 2*(n - 1) = 7 + 2 n - 2 = 5 +2 n[/tex]
S_n = [tex]\frac{a_1 + a_n}{2} * n = 432[/tex]
[tex]\frac{7 + 5 +2 n }{2} *n = 432[/tex]
( 6 + n)* n = 432
n² + 6 n - 432 = 0
Δ = 36 - 4*1*( -432) = 36 + 1728 = 1764
[tex]\sqrt{1764} = 42[/tex]
n = [tex]\frac{- 6 + 42}{2} = 18[/tex]
x = [tex]a_{18} = 5 +2*18 = 41[/tex]
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