Gujarat Boardગુજરાતી માધ્યમધોરણ 11 સાયન્સગણિતદ્રિપદી પ્રમેય1 Mark
MCQ
$2^{2n}-3n-1$ એ ...... વડે વિભાજ્ય છે.
A
$26$
B
$4$
C
$67$
✓
$9$
✓
Answer
Correct option: D.
$9$
D
$2^{2n}-3n-1$
$=4^n-3n-1$
$=(1+3)^n-3n-1$
$=1+\binom{n}{1 }3+\binom{n}{ 2}3^2+...+\binom{n}{ n}3^n-3n-1$
$=1+3n+9\left[\binom{n}{ 2}+...+\binom{n}{ n}3^{n-2}\right]-3n-1$
$=9\left[\binom{n}{ 2}+...+\binom{n}{ n}3^{n-2}\right]$ કે જે $9$ વડે વિભાજ્ય છે.
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