If the 5th term is four times the 4th term and fofour term is six time the 3rd term in (1+x)^n;find the value of n and x

Asked by Jaibir642 | 10th Sep, 2019, 07:58: PM

Expert Answer:

In the expansion of (1+x)n,
left parenthesis 1 plus x right parenthesis to the power of n equals C presubscript 0 presuperscript n plus C presubscript 1 presuperscript n space x plus C presubscript 2 presuperscript n space x squared plus.. plus C presubscript n presuperscript n space x to the power of n
t subscript 5 equals 4 t subscript 4 space a n d space t subscript 4 equals 6 t subscript 3 space... space left parenthesis G i v e n right parenthesis
rightwards double arrow C presubscript 4 presuperscript n space x to the power of 4 equals 4 space C presubscript 3 presuperscript n space x cubed
rightwards double arrow C presubscript 4 presuperscript n space x equals 4 space C presubscript 3 presuperscript n
rightwards double arrow fraction numerator n factorial over denominator left parenthesis n minus 4 right parenthesis factorial 4 factorial end fraction x equals 4 fraction numerator n factorial over denominator left parenthesis n minus 3 right parenthesis factorial 3 factorial end fraction
rightwards double arrow x over 4 equals fraction numerator 4 over denominator n minus 3 end fraction
rightwards double arrow x n minus 3 x equals 16
rightwards double arrow x n equals 3 x plus 16 space.... space left parenthesis i right parenthesis
A n d space t subscript 4 equals 6 t subscript 3
rightwards double arrow C presubscript 3 presuperscript n space x cubed equals 6 space C presubscript 2 presuperscript n space x squared
rightwards double arrow C presubscript 3 presuperscript n space x equals 6 space C presubscript 2 presuperscript n
rightwards double arrow fraction numerator n factorial over denominator left parenthesis n minus 3 right parenthesis factorial 3 factorial end fraction x equals 6 fraction numerator n factorial over denominator left parenthesis n minus 2 right parenthesis factorial 2 factorial end fraction
rightwards double arrow x over 3 equals fraction numerator 6 over denominator n minus 2 end fraction
rightwards double arrow x n minus 2 x equals 18
rightwards double arrow x n equals 2 x plus 18 space.... space left parenthesis i i right parenthesis
U sin g space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis comma space w e space h a v e
x equals 2
S u b s t i t u t i n g space x space i n space left parenthesis i right parenthesis comma space w e space g e t
n equals 11

Answered by Renu Varma | 11th Sep, 2019, 11:22: AM