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# The sequence a1, a2, a3, . . . , an, . . . is such that an = 3an−1 + 2

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Joined: 02 Sep 2009
Posts: 58431
The sequence a1, a2, a3, . . . , an, . . . is such that an = 3an−1 + 2  [#permalink]

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21 Jan 2019, 03:27
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Difficulty:

15% (low)

Question Stats:

95% (01:51) correct 5% (00:54) wrong based on 26 sessions

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The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$a_n = 3a_{n−1} + 2a_{n−2}$$ for all $$n > 2$$. If $$a_3 = 7$$ and $$a_4 = 35$$, what is $$a_2$$?

A 7
B 14
C 17
D 19
E 21

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Re: The sequence a1, a2, a3, . . . , an, . . . is such that an = 3an−1 + 2  [#permalink]

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21 Jan 2019, 06:42
Bunuel wrote:
The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$a_n = 3a_{n−1} + 2a_{n−2}$$ for all $$n > 2$$. If $$a_3 = 7$$ and $$a_4 = 35$$, what is $$a_2$$?

A 7
B 14
C 17
D 19
E 21

if $$a_3 = 7$$ and $$a_4 = 35$$

$$a_3 = 3 a_2 + 2 a_1$$ (a)
$$a_4 = 3 a_3 + 2 a_2$$ (b)

Multiply (a) by 2 and (b) by 3

$$2 a_3 = 6 a_2 + 4 a_1$$
$$3 a_4 = 9 a_3 + 6 a_2$$

Now subtract above equations to get

$$2 a_3 - 3 a_4 = - 9 a_3 + 4 a_1$$
get a relationship as $$11 a_3 = 3 a_4 + 4 a_1$$
$$a_1 = -7$$

$$a_3 = 3 a_2 + 2 a_1$$
$$(a_3 - 2 a_1)/ 3 = 3 a_2$$
$$a_2 = 7$$

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Re: The sequence a1, a2, a3, . . . , an, . . . is such that an = 3an−1 + 2  [#permalink]

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21 Jan 2019, 09:13
Bunuel wrote:
The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$a_n = 3a_{n−1} + 2a_{n−2}$$ for all $$n > 2$$. If $$a_3 = 7$$ and $$a_4 = 35$$, what is $$a_2$$?

A 7
B 14
C 17
D 19
E 21

given expression

an=3 an-1+ 2an-2
a4= 3a3+2a2
solve
a4 & a3 given
2a2= 14
a2= 7
IMO A
Re: The sequence a1, a2, a3, . . . , an, . . . is such that an = 3an−1 + 2   [#permalink] 21 Jan 2019, 09:13
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