SEQUENCES AND SERIES — MATHS STD 11 Science — Question
Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSEQUENCES AND SERIES3 Marks
Question
If a, b, c, d are in G.P., prove that $(a^n + b^n), (b^n + c^n), (c^n + d^n)$ are in G.P.
✓
Answer
Given: a, b, c, d are in G.P.To prove: $(a^n + b^n), (b^n + c^n), (c^n + d^n)$ are in G.P.
$\Rightarrow \frac { b ^ { n } + c ^ { n } } { a ^ { n } + b ^ { n } } = \frac { c ^ { n } + d ^ { n } } { b ^ { n } + c ^ { n } }$
Let $\frac { b } { a } = \frac { c } { b } = \frac { d } { c } = k$
$\therefore \frac { b } { a } = k$
$\Rightarrow$ b = ak
And $\frac { c } { b } = k$
$\Rightarrow c = bk = (ak)k = ak^2$
Also $\frac { d } { c } = k$
$\Rightarrow d = ck = (ak^2)k = ak^3$
Now, $\frac { b ^ { n } + c ^ { n } } { a ^ { n } + b ^ { n } } = \frac { c ^ { n } + d ^ { n } } { b ^ { n } + c ^ { n } }$
$\Rightarrow \frac { ( a k ) ^ { n } + \left( a k ^ { 2 } \right) ^ { n } } { a ^ { n } + ( a k ) ^ { n } } = \frac { \left( a k ^ { 2 } \right) ^ { n } + \left( a k ^ { 3 } \right) ^ { n } } { ( a k ) ^ { n } + \left( a k ^ { 2 } \right) ^ { n } }$
$\Rightarrow \frac { a ^ { n } k ^ { n } + a ^ { n } k ^ { 2 n } } { a ^ { n } + a ^ { n } k ^ { n } } = \frac { a ^ { n } k ^ { 2 n } + a ^ { n } k ^ { 3 n } } { a ^ { n } k ^ { n } + a ^ { n } k ^ { 2 n } }$
$\Rightarrow \frac { a ^ { n } k ^ { n } \left( 1 + k ^ { n } \right) } { a ^ { n } \left( 1 + k ^ { n } \right) } = \frac { a ^ { n } k ^ { 2 n } \left( 1 + k ^ { n } \right) } { a ^ { n } k ^ { n } \left( 1 + k ^ { n } \right) }$
$\Rightarrow k^n = k^n$
Therefore, $(a^n + b^n), (b^n + c^n), (c^n + d^n)$ are in G.P.
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