solve it in details, 3.2^n-4.2^n-2/2^n-2^n+1
Answers
Step-by-step explanation:
•\normalsize\textsf{\underline{As, \: we \: know:-}}
As, we know:-
\normalsize\sf{If \: a \: polynomial \: is \: in \: the \: form \: of \: ax^2 + bx +c \: then,}Ifapolynomialisintheformofax
2
+bx+cthen,
\star⋆ \normalsize\sf{ Sum \: of \: roots \: = \: \frac{-b}{a} \: = \: \frac{-(Coefficient \: of \: x)}{Coefficient \: of \: x^2} }Sumofroots=
a
−b
=
Coefficientofx
2
−(Coefficientofx)
\star⋆ \normalsize\sf{ Product \: of \: roots \: = \: \frac{c}{a} \: = \: \frac{Constant \: term}{Coefficient \: of \: x^2} }Productofroots=
a
c
=
Coefficientofx
2
Constantterm
•\normalsize\textsf{\underline{Now; \: from \: given \: polynomial :-}}
Now; from given polynomial :-
\star⋆ \normalsize\sf{ \: a \: = \: 1}a=1
\star⋆ \normalsize\sf{ \: b \: = \: -7}b=−7
\star⋆ \normalsize\sf{ \: c \: = \: k}c=k
•\normalsize\textsf{\underline{Sum \: of \: products \: (from \: polynomial):-}}
Sum of products (from polynomial):-
➠\normalsize\sf{ a \: + \: b \: = \: \frac{-(-7)}{1} \: = \: 7}a+b=
1
−(−7)
=7
➠{\boxed{\sf{ a \: + \: b \: = \: 7 \: \: \: \: \: ----(eq.1)}}}
a+b=7−−−−(eq.1)
•\normalsize\textsf{\underline{ Product \: of \: roots \: (from \:polynomial):-}}
Product of roots (from polynomial):-
➠\normalsize\sf{ ab \: = \: \frac{k}{1} }ab=
1
k
➠{\boxed{\sf{ ab \: = \: k \: \: \: \: \: -----(eq.2)}}}
ab=k−−−−−(eq.2)
•\normalsize\textsf{\underline{ Also ; \: it \: is \: given \: that:-}}
Also ; it is given that:-
➠{\boxed{\sf{ a \: - \: b \: = \: 1 \: \: \: \: -----(eq.3)}}}
a−b=1−−−−−(eq.3)
•\normalsize\textsf{\underline{ Adding \: equation \: 1 \: and \: 2:-}}
Adding equation 1 and 2:-
\normalsize\sf{ \: \: \: \: \: \: \: \: \: \: \: a \: + \: \cancel{b} \: = \: 7}a+
b
=7
\normalsize\sf{\: \: \: \: \: \: \: \: \: \: a \: - \: \cancel{b} \: = \: 1}a−
b
=1
\normalsize\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2a \: = \: 8}2a=8
➠\normalsize\sf{ a \: = \:\cancel{\frac{8}{2} } }a=
2
8
➠\normalsize\sf{ a \: = \: 4}a=4
➠\large{\boxed{\sf{ a \: = \: 4}}}
a=4
•\normalsize\textsf{\underline{Now; \: put \: the \: value \: in \: equation \: 1:-}}
Now; put the value in equation 1:-
➠\normalsize\sf{ a \: + \: b \: = \: 7}a+b=7
➠\normalsize\sf{ 4 \: + \: b \: = \: 7}4+b=7
➠\normalsize\sf{ b \: = \: 7 \: - \: 4 \: = \: 3}b=7−4=3
➠\large{\boxed{\sf{ b \: = \: 3}}}
b=3
•\normalsize\textsf{\underline{Put \: values \: of \: a \: and \: b \: in \: eq.3:-}}
Put values of a and b in eq.3:-
➠\normalsize\sf{ ab \: = \: k}ab=k
➠\normalsize\sf{ 4 \times\ 3 \: = \: k }4× 3=k
➠\normalsize\sf{ 12 \: = \: k}12=k
➠\normalsize\sf{ k \: = \: 12}k=12
➠\LARGE{\boxed{\sf \orange{ Value \: of \: k \: is \: 12}}}
Valueofkis12