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JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Sequences and Series

For a natural number n, let

a_{n} = 1 9^{n} - 1 2^{n}

. Then, the value of

\frac{31 α _{9} - α _{10}}{57 α _{8}}

is

Answer: 4

Solution

Given

a_{n} = 1 9^{n} - 1 2^{n}

, we need to find

\frac{31 a _{9} - a _{10}}{57 a _{8}}

.

Expanding the numerator:

31 a_{9} - a_{10} = 31 (1 9^{9} - 1 2^{9}) - (1 9^{10} - 1 2^{10})

= 31 \cdot 1 9^{9} - 31 \cdot 1 2^{9} - 1 9^{10} + 1 2^{10}

= 1 9^{9} (31 - 19) - 1 2^{9} (31 - 12)

= 1 9^{9} \cdot 12 - 1 2^{9} \cdot 19

= 12 \cdot 19 (1 9^{8} - 1 2^{8})

= 12 \cdot 19 \cdot a_{8}

Therefore:

\frac{31 a _{9} - a _{10}}{57 a _{8}} = \frac{12 \cdot 19 \cdot a _{8}}{57 a _{8}} = \frac{228}{57} = 4

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Integer answer

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