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{an} is a sequence satisfying an+2 = an+1 + an. If a1 = a4 =

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by Max@Math Revolution » Wed Jan 30, 2019 1:07 am

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[GMAT math practice question]

{an} is a sequence satisfying an+2 = an+1 + an. If a1 = a4 = 2, what is the value of a7?

A. 10
B. 12
C. 14
D. 15
E. 16
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by fskilnik@GMATH » Wed Jan 30, 2019 4:01 am
Max@Math Revolution wrote:[GMAT math practice question]

{an} is a sequence satisfying an+2 = an+1 + an. If a1 = a4 = 2, what is the value of a7?

A. 10
B. 12
C. 14
D. 15
E. 16
$${\rm{sequence}}\,\,:\,\,\,{a_1}\,,\,{a_2}\,,\, \ldots $$
$$\left\{ \matrix{
{a_{n + 2}} = {a_{n + 1}} + {a_n}\,\,\,\left( {n \ge 1} \right) \hfill \cr
{a_1} = {a_4} = 2\,\,\,\,\left( * \right) \hfill \cr} \right.$$`
$$? = {a_7}$$

$$\left\{ \matrix{
\,{a_3} = {a_2} + {a_1} \hfill \cr
\,{a_4} = {a_3} + {a_2} \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,{a_4} = 2{a_2} + {a_1}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{a_2} = 0$$

$$\eqalign{
& {a_3} = {a_2} + {a_1} = 0 + 2 = 2 \cr
& {a_5} = {a_4} + {a_3} = 2 + 2 = 4 \cr
& {a_6} = {a_5} + {a_4} = 4 + 2 = 6 \cr} $$

$$? = {a_7} = {a_6} + {a_5} = 6 + 4 = 10$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Fri Feb 01, 2019 1:34 am
=>

a3 = a2 + a1 = a2 + 2.
a4 = a3 + a2 = a2 + (a2 + 2) = 2a2 + 2 = 2.
Thus, a2 = 0 and a3 = 2.
Therefore,
a5 = a4 + a3 = 2 + 2 = 4,
a6 = a5 + a4 = 4 + 2 = 6,
and
a7 = a6 + a5 = 6 + 4 = 10.

Therefore, the answer is A.
Answer: A
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by Scott@TargetTestPrep » Fri Feb 01, 2019 5:22 pm
Max@Math Revolution wrote:[GMAT math practice question]

{an} is a sequence satisfying an+2 = an+1 + an. If a1 = a4 = 2, what is the value of a7?

A. 10
B. 12
C. 14
D. 15
E. 16
Starting with a3, we have:

a3 = a2 + a1 → a3 = a2 + 2

Next, we have a4:

a4 = a3 + a2

We were given that a4 = 2. Substituting, we have:

2 = a2 + 2 + a2

0 = 2a2

0 = a2

Since a2 = 0, a3 = 2. Now we have:

a5 = a4 + a3

a5 = 2 + 2 = 4

Then we have:

a6 = a5 + a4

a6 = 4 + 2 = 6

Lastly, we have:

a7 = a6 + a5

a7 = 6 + 4 = 10

Answer: A

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