If , then is equal to: [2026]

Question

If $26 (\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})) = 3^{13} - α$ , then $α$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

$26 [\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})] = 3^{13} - α$

$= 2 [2^{3} \cdot {}^{13}C_{3} + 2^{5} \cdot {}^{13}C_{5} + 2^{7} \cdot {}^{13}C_{7} + . . . + 2^{13} \cdot {}^{13}C_{13}]$

$= 2 [\frac{{(1 + 2)}^{13} - {(1 - 2)}^{13}}{2} - 26]$

$= 3^{13} + 1 - 52 = 3^{13} - 51 ∴ α = 51$

Solution

Q.
If $26 (\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})) = 3^{13} - α$ , then $α$ is equal to: [2026]

1 45

2 48

3 51

4 54

Want Us to Email you About Special Offers & Updates?