Which term of 0.0004,0.02,0.1,……………… is 12 ? A) 7 B) 5

1.	Which term of 0.0004,0.02,0.1,……………… is 12 ? A) 7 B) 5 C) 8 D) 6
Answer» Correct option is (D) 6 Let \(a_1=0.004=\frac4{1000}\) \(a_2=0.02=\frac2{100}\) \(a_3=0.1=\frac1{10}\) Now, \(\frac{a_2}{a_1}=\cfrac{\frac{2}{100}}{\frac{4}{1000}}\) \(=\frac{2}{100}\times\frac{1000}{4}\) \(=\frac{10}{2}=5\) \(\frac{a_3}{a_2}=\cfrac{\frac{1}{10}}{\frac{2}{100}}\) \(=\frac{1}{10}\times\frac{100}{2}\) \(=\frac{10}{2}=5\) Hence, 0.004, 0.02, 0.1, ......… is a geometric progression whose first term is \(a=a_1=0.004=\frac4{1000}\) & common ratio \(=r=\frac{a_2}{a_1}=5\) Let \(n^{th}\) term of the G.P. is 12.5. i.e., \(a_n=12.5\) \(\Rightarrow ar^{n-1}=12.5\) \(\Rightarrow\frac4{1000}\times5^{n-1}=\frac{125}{10}\) \(\Rightarrow5^{n-1}=\frac{125}{10}\times\frac{1000}{4}\) \(=125\times25\) \(=5^3\times5^2=5^5\) \(\therefore n-1=5\) \(\Rightarrow n=5+1=6\) Hence, \(6^{th}\) term of given sequence is 6. Correct option is D) 6

Which term of 0.0004,0.02,0.1,……………… is 12 ? A) 7 B) 5 C) 8 D) 6

Answer»

Correct option is (D) 6

Let \(a_1=0.004=\frac4{1000}\)

\(a_2=0.02=\frac2{100}\)

\(a_3=0.1=\frac1{10}\)

Now, \(\frac{a_2}{a_1}=\cfrac{\frac{2}{100}}{\frac{4}{1000}}\)

\(=\frac{2}{100}\times\frac{1000}{4}\)

\(=\frac{10}{2}=5\)

\(\frac{a_3}{a_2}=\cfrac{\frac{1}{10}}{\frac{2}{100}}\)

\(=\frac{1}{10}\times\frac{100}{2}\)

\(=\frac{10}{2}=5\)

Hence, 0.004, 0.02, 0.1, ......… is a geometric progression whose first term is \(a=a_1=0.004=\frac4{1000}\)

& common ratio \(=r=\frac{a_2}{a_1}=5\)

Let \(n^{th}\) term of the G.P. is 12.5.

i.e., \(a_n=12.5\)

\(\Rightarrow ar^{n-1}=12.5\)

\(\Rightarrow\frac4{1000}\times5^{n-1}=\frac{125}{10}\)

\(\Rightarrow5^{n-1}=\frac{125}{10}\times\frac{1000}{4}\)

\(=125\times25\)

\(=5^3\times5^2=5^5\)

\(\therefore n-1=5\)

\(\Rightarrow n=5+1=6\)

Hence, \(6^{th}\) term of given sequence is 6.

Correct option is D) 6