11)
5, -5, -15, -25, ...
-5-5 = -10; -15-(-5) = -10; -25-(-15) = -10 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 10 less than previous one)
difference: d = -10
first term: a = 5
so the formula:
[tex]a_n=a+d(n-1)\\\\a_n=5+(-10)(n-1)\\\\a_n =5-10n+10\\\\\underline{a_n=-10n+15}[/tex]
and:
[tex]a_{20}=-10\cdot20+15=-200+15=-185[/tex]
12)
19, 26, 33, 40, ...
26-19 = 7; 33-26 = 7; 40-33 = 7 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 7 more than previous one)
difference: d = 7
first term: a = 19
so the formula:
[tex]a_n=a+d(n-1)\\\\a_n=19+7(n-1)\\\\a_n =19+7n-7\\\\\underline{a_n=7n+12}[/tex]
and:
[tex]a_{39}=7\cdot39+12=273+12=285[/tex]
13)
-20, -29, -38, -47, ...
-29-(-20) = -9; -38-(-28) = -9; -47-(-38) = -9 ⇒ this is an example of arithmetic sequence that has explicit patern (evey next term is 9 less than previous one)
difference: d = -9
first term: a = -20
so the formula:
[tex]a_n=a+d(n-1)\\\\a_n=-20+(-9)(n-1)\\\\a_n =-20-9n+9\\\\\underline{a_n=-9n-11}[/tex]
and:
[tex]a_{12}=-9\cdot12-11=-108-11=-119[/tex]
11)
1, -2, 3, -4, ...
-5-1 = -3; 3-(-2) = 5 ≠ -3 ⇒ this is NOT an example of arithmetic sequence
however it has explicit patern:
[tex]a_1=1 =1\cdot\left(-1\right)^0= 1\cdot\left(-1\right)^{1-1}\\\\a_2=-2=2\cdot\left(-1\right)^1=2\cdot\left(-1\right)^{2-1}\\\\a_3=3=3\cdot\left(-1\right)^2=3\cdot\left(-1\right)^{3-1}\\\\a_4=-4=4\cdot\left(-1\right)^3=4\cdot\left(-1\right)^{4-1}\\\\\underline{a_n=n\cdot\left(-1\right)^{n-1}}[/tex]
so:
[tex]a_{33}=33\cdot\left(-1\right)^{33-1}=33\cdot\left(-1\right)^{32}=33\cdot1=33[/tex]
