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8. Sequences and Series
easy
If $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}}$ be the $A.M.$ of $a$ and $b$, then $n=$
A
$1$
B
$- 1$
C
$0$
D
None of these
Solution
(c) $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}} = \frac{{a + b}}{2}$
$ \Rightarrow $ ${a^{n + 1}} – a{b^n} + {b^{n + 1}} – b{a^n} = 0$
$ \Rightarrow $$(a – b)({a^n} – {b^n}) = 0$
If ${a^n} – {b^n} = 0$.
Then ${\left( {\frac{a}{b}} \right)^n} = 1 = {\left( {\frac{a}{b}} \right)^0}$.
Hence $n = 0$.
Standard 11
Mathematics