Answer
Verified
423.6k+ views
Hint: In the questions we have to find the common ratio and then using the formula of the nth term of GP we can find which position the term 256 holds by equating it.
Complete step-by-step answer:
In the given series let's name the terms first i.e. \[{a_1}\] = 2, \[{a_2}\] =\[2\sqrt 2 \], \[{a_3}\]= 4
Let 256 be the \[{a_n}\]th term
Now, the common ratio \[r\] is given by
\[r = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{{a_3}}}{{{a_2}}} = .......... = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\]
∴ \[r\]=\[\dfrac{{2\sqrt 2 }}{2}\]=\[\sqrt 2 \]
Using the formula of nth term
\[{a_n} = a{r^{n - 1}}\]
\[ \Rightarrow 256 = 2{\left( {\sqrt 2 } \right)^{n - 1}}\]
\[ \Rightarrow 128 = {\left( {\sqrt 2 } \right)^{n - 1}}\]
Converting 128 in terms of power of 2
\[ \Rightarrow {2^7} = {2^{\dfrac{{n - 1}}{2}}}\]
Since, bases are equal, therefore powers can also be equated.
\[ \Rightarrow 7 = \dfrac{{n - 1}}{2}\]
\[ \Rightarrow 14 = n - 1\]
\[\therefore n = 15\]
∴ 256 is the 15th term.
Note: Geometric progression is a sequence of numbers where each new term after the first is obtained by multiplying the preceding term by a constant r called common ratio. The common ratio is given by the formula \[r = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{{a_3}}}{{{a_2}}} = .......... = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\].
Complete step-by-step answer:
In the given series let's name the terms first i.e. \[{a_1}\] = 2, \[{a_2}\] =\[2\sqrt 2 \], \[{a_3}\]= 4
Let 256 be the \[{a_n}\]th term
Now, the common ratio \[r\] is given by
\[r = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{{a_3}}}{{{a_2}}} = .......... = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\]
∴ \[r\]=\[\dfrac{{2\sqrt 2 }}{2}\]=\[\sqrt 2 \]
Using the formula of nth term
\[{a_n} = a{r^{n - 1}}\]
\[ \Rightarrow 256 = 2{\left( {\sqrt 2 } \right)^{n - 1}}\]
\[ \Rightarrow 128 = {\left( {\sqrt 2 } \right)^{n - 1}}\]
Converting 128 in terms of power of 2
\[ \Rightarrow {2^7} = {2^{\dfrac{{n - 1}}{2}}}\]
Since, bases are equal, therefore powers can also be equated.
\[ \Rightarrow 7 = \dfrac{{n - 1}}{2}\]
\[ \Rightarrow 14 = n - 1\]
\[\therefore n = 15\]
∴ 256 is the 15th term.
Note: Geometric progression is a sequence of numbers where each new term after the first is obtained by multiplying the preceding term by a constant r called common ratio. The common ratio is given by the formula \[r = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{{a_3}}}{{{a_2}}} = .......... = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\].
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE