divide 54 is in four parts in ap such that the ratio of product of their extrmes (1st and 4th) to the product of meas (2nd and 3rd in 5:6
Answers
Step-by-step explanation:
Given:
↬ 56 is divided into four parts which form an A.P.
↬ Ratio of the product of extremes of given A.P. to the product of their means is 5:6
To Find:
All the parts of 56
Things to know before solving question
(a+b)(a-b)= a² -b²
Solution:
Let the four parts be a-3d, a-d, a+d, a+3d such that they are in A.P.
Now,
Sum of all parts= 56
⇒ a-3d+a-d+a+d+a+3d= 56
⇒ 4a= 56
⇒ a= 14
Now,
According to the question,
\sf{\dfrac{Product\:of\:extremes\:of\:A.P.}{Product\:of\:means\:of\:A.P.}=\dfrac{5}{6}}
ProductofmeansofA.P.
ProductofextremesofA.P.
=
6
5
i.e. \sf{\dfrac{Product\:of\:first\:and\:fourth\:terms\:of\:A.P.}{Product\:of\:second\:and\:third\:terms\:of\:A.P.}=\dfrac{5}{6}}
ProductofsecondandthirdtermsofA.P.
ProductoffirstandfourthtermsofA.P.
=
6
5
\longrightarrow\sf{\dfrac{(a-3d)(a+3d)}{(a-d)(a+d)}=\dfrac{5}{6}}⟶
(a−d)(a+d)
(a−3d)(a+3d)
=
6
5
\longrightarrow\sf{\dfrac{a^{2}-(3d)^{2}}{a^{2}-d^{2}}=\dfrac{5}{6}}⟶
a
2
−d
2
a
2
−(3d)
2
=
6
5
\longrightarrow\sf{\dfrac{a^{2}-9d^{2}}{a^{2}-d^{2}}=\dfrac{5}{6}}⟶
a
2
−d
2
a
2
−9d
2
=
6
5
\longrightarrow\sf{6(a^{2}-9d^{2})=5(a^{2}-d^{2})}⟶6(a
2
−9d
2
)=5(a
2
−d
2
)
\longrightarrow\sf{6a^{2}-54d^{2}=5a^{2}-5d^{2}}⟶6a
2
−54d
2
=5a
2
−5d
2
\longrightarrow\sf{6a^{2}-5a^{2}=54d^{2}-5d^{2}}⟶6a
2
−5a
2
=54d
2
−5d
2
\longrightarrow\sf{a^{2}=49d^{2}}⟶a
2
=49d
2
\longrightarrow\sf{49d^{2}=a^{2}}⟶49d
2
=a
2
On putting value of a in above equation, we get
\longrightarrow\sf{49d^{2}=14^{2}}⟶49d
2
=14
2
\longrightarrow\sf{49d^{2}=196}⟶49d
2
=196
\longrightarrow\sf{d^{2}=\dfrac{\cancel{196}}{\cancel{49}}}⟶d
2
=
49
196
\longrightarrow\sf{d^{2}=2}⟶d
2
=2
\longrightarrow\sf{d=\pm2}⟶d=±2
Now,
Case-1 ,when d=2
First part= a-3d= 14-3(2)= 8
Second part= a-d= 14-2= 12
Third part= a+d= 14+2= 16
Fourth part= a+3d= 14+3(2)= 20
Case-2 ,when d= -2
First part= a-3d= 14-3(-2)= 20
Second part= a-d= 14-(-2)= 16
Third part= a+d= 14+(-2)= 12
Fourth part= a+3d= 14+3(-2)= 8
Hence, all four parts are 8, 12, 16 and 20
Answer:
8,12,16,20
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