JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

S = 109 + \frac{108}{5} + \frac{107}{5^2} + \dots + \frac{2}{5^{107}} + \frac{1}{5^{108}}

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Explanation

Step 1: Write the original equation ( $S$ )

S = 109 + \frac{108}{5} + \frac{107}{5^2} + \dots + \frac{2}{5^{107}} + \frac{1}{5^{108}} \quad \text{---(i)}

Step 2: Multiply $S$ by $1/5$ and shift by one term

\frac{1}{5}S = \frac{109}{5} + \frac{108}{5^2} + \dots + \frac{3}{5^{107}} + \frac{2}{5^{108}} + \frac{1}{5^{109}} \quad \text{---(ii)}

Step 3: Subtract (ii) from (i)

S - \frac{1}{5}S = 109 + \left( \frac{108-109}{5} \right) + \left( \frac{107-108}{5^2} \right) + \dots + \left( \frac{1-2}{5^{108}} \right) - \frac{1}{5^{109}}

\frac{4}{5}S = 109 - \left[ \frac{1}{5} + \frac{1}{5^2} + \dots + \frac{1}{5^{108}} \right] - \frac{1}{5^{109}}

Step 4: Use the formula for Geometric Series sum

Bracket ke andar wali series ek GP hai jisme $a = 1/5$ , $r = 1/5$ , aur $n = 108$ terms hain.

\text{Sum} = \frac{\frac{1}{5} \left( 1 - \frac{1}{5^{108}} \right)}{1 - \frac{1}{5}} = \frac{1}{4} \left( 1 - \frac{1}{5^{108}} \right)

Substututing this back:

\frac{4}{5}S = 109 - \frac{1}{4} + \frac{1}{4 \cdot 5^{108}} - \frac{1}{5^{109}}

\frac{4}{5}S = \frac{435}{4} + \frac{5 - 4}{4 \cdot 5^{109}} = \frac{435}{4} + \frac{1}{4 \cdot 5^{109}}

Step 5: Multiply by $4 \times 5$ (which is $20$ ) to find $16S$

Multiplying the whole equation by $20$ :

16S = 435 \cdot 5 + \frac{5}{5^{109}} = 2175 + \frac{1}{5^{108}}

Step 6: Final Calculation

Hame nikalna hai: $16S - (25)^{-54}$ .

Note karein ki $(25)^{-54} = (5^2)^{-54} = 5^{-108} = \frac{1}{5^{108}}$ .

16S - \frac{1}{5^{108}} = \left( 2175 + \frac{1}{5^{108}} \right) - \frac{1}{5^{108}}

= 2175

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