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Question Number 8031 by Nayon last updated on 28/Sep/16

(√2) ≈((19601)/(13860))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\sqrt{\mathrm{2}}\:\approx\frac{\mathrm{19601}}{\mathrm{13860}} \\ $$

Answered by prakash jain last updated on 28/Sep/16

(√2)=1+(1/(1+(√2))) ...(i) putting value of (√2) from (i) in RHS of (i) ⇒(√2)=1+(1/(1+1+(1/(√2))))=(1/(2+(1/(√2)))) ⇒(√2)=1+(1/(2+(1/(2+(1/(√2)))))) and so on (√2)=1+(1/(2+(1/(2+(1/(2+⋱)))))) approxmial value calculations (√2)≈1+(1/2)=(3/2) (√2)≈1+(1/(2+1/2))=1+(2/5)=(7/5) continuing this way (√2)≈1+(1/(1+(√2)))=1+(1/(1+7/5))=((17)/(12)) (√2)≈1+(1/(1+17/12))=((41)/(29)) continuing this way will result in more accurate values of (√2) ((99)/(70)),((239)/(169)),((577)/(408)),((1393)/(985)),((3363)/(2379)),((8119)/(5741)),((19601)/(13860))

$$\sqrt{\mathrm{2}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}}}\:\:\:\:...\left({i}\right) \\ $$$$\mathrm{putting}\:\mathrm{value}\:\mathrm{of}\:\sqrt{\mathrm{2}}\:\mathrm{from}\:\left({i}\right)\:\mathrm{in}\:\mathrm{RHS} \\ $$$$\mathrm{of}\:\left({i}\right) \\ $$$$\Rightarrow\sqrt{\mathrm{2}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}=\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}} \\ $$$$\Rightarrow\sqrt{\mathrm{2}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}}} \\ $$$$\mathrm{and}\:\mathrm{so}\:\mathrm{on} \\ $$$$\sqrt{\mathrm{2}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{2}+\ddots}}} \\ $$$$\mathrm{approxmial}\:\mathrm{value}\:\mathrm{calculations} \\ $$$$\sqrt{\mathrm{2}}\approx\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\sqrt{\mathrm{2}}\approx\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}+\mathrm{1}/\mathrm{2}}=\mathrm{1}+\frac{\mathrm{2}}{\mathrm{5}}=\frac{\mathrm{7}}{\mathrm{5}} \\ $$$$\mathrm{continuing}\:\mathrm{this}\:\mathrm{way} \\ $$$$\sqrt{\mathrm{2}}\approx\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}}}=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{7}/\mathrm{5}}=\frac{\mathrm{17}}{\mathrm{12}} \\ $$$$\sqrt{\mathrm{2}}\approx\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{17}/\mathrm{12}}=\frac{\mathrm{41}}{\mathrm{29}} \\ $$$$\mathrm{continuing}\:\mathrm{this}\:\mathrm{way}\:\mathrm{will}\:\mathrm{result} \\ $$$$\mathrm{in}\:\mathrm{more}\:\mathrm{accurate}\:\mathrm{values}\:\mathrm{of}\:\sqrt{\mathrm{2}} \\ $$$$\frac{\mathrm{99}}{\mathrm{70}},\frac{\mathrm{239}}{\mathrm{169}},\frac{\mathrm{577}}{\mathrm{408}},\frac{\mathrm{1393}}{\mathrm{985}},\frac{\mathrm{3363}}{\mathrm{2379}},\frac{\mathrm{8119}}{\mathrm{5741}},\frac{\mathrm{19601}}{\mathrm{13860}} \\ $$

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