JEE Advance - Mathematics (2012 - Paper 2 Offline - No. 15)
Which of the following is correct?
$${a_{17}} = {a_{16}} + {a_{15}}$$
$${c_{17}} \ne {c_{16}} + {c_{15}}$$
$${b_{17}} \ne {b_{16}} + {c_{16}}$$
$${a_{17}} = {c_{17}} + {b_{16}}$$
Explanation
For $$a_n$$
Case I : If the unit digit is 1, and rest $$(n-1)$$ places are filled as $$a_{n-1}$$
$$\mathrm{\underbrace {1\underline {} \,\underline {} \,\underline {} \,\underline {...} \,\underline {...} \,\underline {} }_{(n - 1)\,places}\underline 1}$$
Case II : If the unit digit is 0, then tenth place must be 1 and rest $$(n-2)$$ places are filled as $$a_{n-2}$$
$$\mathrm{\underbrace {1\underline {} \,\underline {} \,\underline {} \,\underline {} \,....\,\underline {} }_{(n - 2)\,place}\underline 1 \underline 0}$$
$$\begin{aligned} & \text { Hence, } a_n =a_{n-1}+a_{n-2} \\ \Rightarrow \quad & a_{17} =a_{16}+a_{15} \end{aligned}$$
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