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Find the first five terms of the recursive sequence. Show all your work. an = 3a n-1 - 6 where a1 = 7

Sagot :

Answer:  7, 15, 39, 111, 327

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Work Shown:

First replace every copy of n with 2

[tex]a_n = 3*(a_{n-1}) - 6\\\\a_2 = 3*(a_{2-1}) - 6\\\\a_2 = 3*(a_{1}) - 6\\\\a_2 = 3*(7) - 6\\\\a_2 = 21 - 6\\\\a_2 = 15\\\\[/tex]

Notice how the second term [tex]a_2[/tex] relies on the first term [tex]a_1[/tex]

Then repeat for n = 3

[tex]a_n = 3*(a_{n-1}) - 6\\\\a_3 = 3*(a_{3-1}) - 6\\\\a_3 = 3*(a_{2}) - 6\\\\a_3 = 3*(15) - 6\\\\a_3 = 45 - 6\\\\a_3 = 39\\\\[/tex]

Same goes with n = 4

[tex]a_n = 3*(a_{n-1}) - 6\\\\a_4 = 3*(a_{4-1}) - 6\\\\a_4 = 3*(a_{3}) - 6\\\\a_4 = 3*(39) - 6\\\\a_4 = 117 - 6\\\\a_4 = 111\\\\[/tex]

Finally plug in n = 5

[tex]a_n = 3*(a_{n-1}) - 6\\\\a_5 = 3*(a_{5-1}) - 6\\\\a_5 = 3*(a_{4}) - 6\\\\a_5 = 3*(111) - 6\\\\a_5 = 333 - 6\\\\a_5 = 327\\\\[/tex]

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We have this summary:

[tex]a_1 = 7\\\\a_2 = 15\\\\a_3 = 39\\\\a_4 = 111\\\\a_5 = 327\\\\[/tex]

The first five terms are:  7, 15, 39, 111, 327