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Question Number 24937 by kosarrr last updated on 29/Nov/17

Commented by behi.8.3.4.17@gmail.com last updated on 29/Nov/17

2^(k+1) =3×2^(k−1) +16  2×2^k =(3/2)×2^k +16  4×2^k =3×2^k +2×16  2^k =2^5 ⇒k=5   .■  .............  2^(2k) =56+2^k ⇒(2^k )^2 −2^k −56=0  ⇒(2^k −8)(2^k +7)=0⇒2^k =8=2^3 ⇒k=3 .■

$$\mathrm{2}^{\mathrm{k}+\mathrm{1}} =\mathrm{3}×\mathrm{2}^{\mathrm{k}−\mathrm{1}} +\mathrm{16} \\ $$$$\mathrm{2}×\mathrm{2}^{\mathrm{k}} =\frac{\mathrm{3}}{\mathrm{2}}×\mathrm{2}^{\mathrm{k}} +\mathrm{16} \\ $$$$\mathrm{4}×\mathrm{2}^{\mathrm{k}} =\mathrm{3}×\mathrm{2}^{\mathrm{k}} +\mathrm{2}×\mathrm{16} \\ $$$$\mathrm{2}^{\mathrm{k}} =\mathrm{2}^{\mathrm{5}} \Rightarrow\boldsymbol{{k}}=\mathrm{5}\:\:\:.\blacksquare \\ $$$$............. \\ $$$$\mathrm{2}^{\mathrm{2}\boldsymbol{{k}}} =\mathrm{56}+\mathrm{2}^{\boldsymbol{{k}}} \Rightarrow\left(\mathrm{2}^{\mathrm{k}} \right)^{\mathrm{2}} −\mathrm{2}^{\mathrm{k}} −\mathrm{56}=\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{2}^{\mathrm{k}} −\mathrm{8}\right)\left(\mathrm{2}^{\mathrm{k}} +\mathrm{7}\right)=\mathrm{0}\Rightarrow\mathrm{2}^{\mathrm{k}} =\mathrm{8}=\mathrm{2}^{\mathrm{3}} \Rightarrow\boldsymbol{{k}}=\mathrm{3}\:.\blacksquare \\ $$

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