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Arithmetic Progression

A sequence of...

Question

A sequence of numbers $a_{1}, a_{2}, a_{3}, . . . . .$ satisfies the relation $a_{n + 1} = a_{n} + a_{n - 1}$ for $n \geq 2$ . Find $a_{4}$ given $a_{1} = a_{2} = 1$ .

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Solution

A sequence of numbers $a_{1}, a_{2}, a_{3} - - -$ satisfies the relation $a_{n + 1} = a_{n} + a_{n - 1} n \geq 2$
Given, $a_{1} = a_{2} = 1$
Now, $a_{n + 1} = a_{n} + a_{n - 1}$
for $n = 2 \Rightarrow a_{3} = a_{2} + a_{1} ∴ a_{3} = 2$ [Putting the value of $a_{1}, a_{2}$ ]
Now, $a_{4} = a_{3} + a_{2} ∴ a_{4} = 2 + 1 ∴ a_{4} = 3$

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