Find the rational numbers having the following decimal expansions :3.5

1.	Find the rational numbers having the following decimal expansions :3.5\(\bar 2\)
Answer» x = 3.522222222 ….. x = 3.5+0.02 + 0.002 + 0.0002 + …∞ ⇒ x = 3.5+2(0.01 + 0.001 + 0.0001 + …∞ ) ⇒ x = 3.5 + 2 \(\bigg(\frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + ...... \infty\bigg)\) ⇒ x = 3.5 + 2S Where S = \(\frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + ...... \infty\) We observe that the above progression possess a common ratio. So it is a geometric progression. Common ratio = 1/10 and first term (a) = 1/100 Sum of infinite GP = \(\frac{a}{1-k}\), where a is the first term and k is the common ratio. Note: We can only use the above formula if \|k\|<1 ∴ we can use the formula for the sum of infinite GP. ⇒ S = \(\cfrac{\frac{1}{100}}{1-(\frac{1}{10})}\) = \(\frac{1}{90}\) = \(\frac{1}{90}\) ∴ x = 3.5 + 2(1/90) ⇒ x = (35/10) + 1/45 = (315+2)/90 = 317/90

Find the rational numbers having the following decimal expansions :3.5\(\bar 2\)

Answer»

x = 3.522222222 …..

x = 3.5+0.02 + 0.002 + 0.0002 + …∞

⇒ x = 3.5+2(0.01 + 0.001 + 0.0001 + …∞ )

⇒ x = 3.5 + 2 \(\bigg(\frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + ...... \infty\bigg)\)

⇒ x = 3.5 + 2S

Where S = \(\frac{1}{100} + \frac{1}{1000} + \frac{1}{10000} + ...... \infty\)

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = 1/10 and first term (a) = 1/100

Sum of infinite GP = \(\frac{a}{1-k}\), where a is the first term and k is the common ratio.

Note: We can only use the above formula if |k|<1

∴ we can use the formula for the sum of infinite GP.

⇒ S = \(\cfrac{\frac{1}{100}}{1-(\frac{1}{10})}\) = \(\frac{1}{90}\) = \(\frac{1}{90}\)

∴ x = 3.5 + 2(1/90)

⇒ x = (35/10) + 1/45 = (315+2)/90 = 317/90