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Michael Sipser Edition 3 Exercise 1 Question 61 (Page No. 92)

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Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}=\sum^{*}a\sum^{k-1}.$ Prove that for each $k,$ $\text{no DFA}$ can recognize $C_{k}$ with fewer than $2^{k}$ states.
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