A sequence is given by the rule a_n+2= 2a_n+1 +a_n, where n>0. Also, a_1=1 and a_2=2. Find the value of a_100*a_101?

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705-805

Sequences

A sequence is given by the rule $a_{n+2}=\frac{2}{a_{n+1}} +a_n$ , where n>0. Also, $a_1$ =1 and $a_2$ =2. Find the value of $a_{100}*a_{101}$ ?

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240

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Official Explanation

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We have

a_{n+2} = \frac{2}{a_{n+1}} + a_n

.

Rearranging:

a_{n+2} \cdot a_{n+1} = 2 + a_n \cdot a_{n+1}

.

Let

P_n = a_n \cdot a_{n+1}

. Then the recurrence becomes

P_{n+1} = 2 + P_n

.

We know

P_1 = a_1 \cdot a_2 = 1 \cdot 2 = 2

.

Thus

P_2 = 2 + P_1 = 2 + 2 = 4

P_3 = 2 + P_2 = 6

, etc.…

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What this question tests

What it tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Common trap

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.

A sequence is given by the rule an+2=2an+1+ana_{n+2}=\frac{2}{a_{n+1}} +a_nan+2​=an+1​2​+an​, where n>0. Also, a1a_1a1​=1 and a2a_2a2​=2. Find the value of a100∗a101a_{100}*a_{101}a100​∗a101​?