One of the 5 geometric means between 1/3 and 243 is A) 81 B) 80 C)

1.	One of the 5 geometric means between 1/3 and 243 is A) 81 B) 80 C) 79 D) 82
Answer» Correct option is (A) 81 Let \(ar,ar^2,ar^3,ar^4,ar^5\) are 5 geometric means between \(\frac{1}{3}\) and 243. \(\therefore\) \(\frac13,ar,ar^2,ar^3,ar^4,ar^5,243\) will form a G.P. \(\therefore a=\frac13\;\&\;ar^6=243\) \(\Rightarrow r^6=\frac{243}a=\frac{243}{\frac13}\) \(=243\times3=729\) \(\Rightarrow r^6=3^6\) \(\Rightarrow r=3\) \(\therefore\) \(ar=\frac13\times3=1,\) \(ar^2=\frac13\times9=3,\) \(ar^3=\frac13\times27=9,\) \(ar^4=\frac13\times81=27,\) \(ar^5=\frac13\times243=81\) Hence, 1, 3, 9, 27 and 81 are 5 geometric means. Hence, one of the 5 geometric mean is 81. Correct option is A) 81

One of the 5 geometric means between 1/3 and 243 is A) 81 B) 80 C) 79 D) 82

Answer»

Correct option is (A) 81

Let \(ar,ar^2,ar^3,ar^4,ar^5\) are 5 geometric means between \(\frac{1}{3}\) and 243.

\(\therefore\) \(\frac13,ar,ar^2,ar^3,ar^4,ar^5,243\) will form a G.P.

\(\therefore a=\frac13\;\&\;ar^6=243\)

\(\Rightarrow r^6=\frac{243}a=\frac{243}{\frac13}\)

\(=243\times3=729\)

\(\Rightarrow r^6=3^6\)

\(\Rightarrow r=3\)

\(\therefore\) \(ar=\frac13\times3=1,\)

\(ar^2=\frac13\times9=3,\)

\(ar^3=\frac13\times27=9,\)

\(ar^4=\frac13\times81=27,\)

\(ar^5=\frac13\times243=81\)

Hence, 1, 3, 9, 27 and 81 are 5 geometric means.

Hence, one of the 5 geometric mean is 81.

Correct option is A) 81