[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
{an} is a sequence satisfying an+2 = an+1 + an. If a1 = a4 =
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- Max@Math Revolution
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$${\rm{sequence}}\,\,:\,\,\,{a_1}\,,\,{a_2}\,,\, \ldots $$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
$$\left\{ \matrix{
{a_{n + 2}} = {a_{n + 1}} + {a_n}\,\,\,\left( {n \ge 1} \right) \hfill \cr
{a_1} = {a_4} = 2\,\,\,\,\left( * \right) \hfill \cr} \right.$$`
$$? = {a_7}$$
$$\left\{ \matrix{
\,{a_3} = {a_2} + {a_1} \hfill \cr
\,{a_4} = {a_3} + {a_2} \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,{a_4} = 2{a_2} + {a_1}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{a_2} = 0$$
$$\eqalign{
& {a_3} = {a_2} + {a_1} = 0 + 2 = 2 \cr
& {a_5} = {a_4} + {a_3} = 2 + 2 = 4 \cr
& {a_6} = {a_5} + {a_4} = 4 + 2 = 6 \cr} $$
$$? = {a_7} = {a_6} + {a_5} = 6 + 4 = 10$$
We follow the notations and rationale taught in the GMATH method.
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Fabio.
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- Max@Math Revolution
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=>
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
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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Starting with a3, we have: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
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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