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1) Find the mean terms between 3 and 27, if they are in continued proportion?
2) If the sum of four consecutive of odd numbers is 96. Find the numbers.
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Step-by-step explanation:
[tex]\tt \large \: {\red{ \underline {question \: (1) : }}}[/tex]
[tex]\tt \large \: {\pink{ \underline {solution: }}} \\ [/tex]
suppose the mean term be x .
therefore, 3 , x , 27 are in continued proportion.
then,
[tex] \tt \: 3 \: \ratio \: x \: = \: x \: \ratio \: 27 \\ \\ \because \sf \: product \: of \: means = product \: of \: extremes \\ \\ \therefore \sf x \times x = 27 \times 3 \\ \\ \sf \: {x}^{2} = 81 \\ \\ \sf \: x = \sqrt{81} \\ \\ \boxed{ \sf x = 9} \\ [/tex]
Hence, mean term is 9 .
[tex] \\ \rule{190pt}{2pt} \\ [/tex]
[tex]\tt \large \: {\red{ \underline {question \: (2) : }}} \\ [/tex]
[tex]\tt \large \: {\blue{ \underline {given : }}} \\ [/tex]
[tex] \\ \tt \large \: {\pink{ \underline {to \: find: }}} \\ [/tex]
[tex] \\ \tt \large \: {\pink{ \underline {solution \: : }}} \\ [/tex]
[tex] \\ \underline{\bf acc \: to \: the \: question} \\ \\ \sf \: (x + 1) + (x + 3) + (x + 5) + (x +7 ) = 96 \\ \\ \sf \: 4x + 16 = 96 \\ \\ \sf4x = 96 - 16\\ \\ \sf \: 4x = 80 \\ \\ \boxed{\sf x = 20} \\ [/tex]
Therefore,
Numbers are :- 21 , 23 , 25 , 27 .
[tex] \\ [/tex]
Verified answer
Answer:
[tex]\:\boxed{\begin{aligned}& \:\sf \:(1). \: 9 \: is \: mean \: continued \: proportion \: between \: 3 \: and \: 27 \: \\ \\& \:\sf \: (2). \: 4 \: consective \: odd \: numbers \: are \: 21, \: 23, \: 25, \: 27\end{aligned}} \qquad \: \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Let assume that x be the mean continued proportion between 3 and 27.
So, by definition of mean continued proportion, we have
[tex]\sf \: 3 : x :: x : 27 \\ \\ [/tex]
[tex]\sf \: \dfrac{3}{x} = \dfrac{x}{27} \\ \\ [/tex]
[tex]\sf \: {x}^{2} = 3 \times 27 \\ \\ [/tex]
[tex]\sf \: x = \sqrt{3 \times 27} \\ \\ [/tex]
[tex]\sf \: x = \sqrt{3 \times 3 \times 3 \times 3} \\ \\ [/tex]
[tex]\sf \: x = 3 \times 3 \\ \\ [/tex]
[tex]\sf\implies \bf \: x = 9 \\ \\ [/tex]
Hence,
[tex]\sf\implies \bf \: 9 \: is \: mean \: continued \: proportion \: between \: 3 \: and \: 27 \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Let assume that 4 consecutive odd numbers be x, x + 2, x + 4, x + 6 respectively.
According to statement, sum of four consecutive odd numbers is 96.
[tex]\sf \: x + x + 2 + x + 4 + x + 6 = 96 \\ \\ [/tex]
[tex]\sf \: 4x + 12 = 96 \\ \\ [/tex]
[tex]\sf \: 4x = 96 - 12\\ \\ [/tex]
[tex]\sf \: 4x = 84\\ \\ [/tex]
[tex]\sf\implies \sf \: x = 21\\ \\ [/tex]
Thus,
[tex]\bf\implies \: 4 \: consective \: odd \: numbers \: are \: 21, \: 23, \: 25, \: 27 \\ \\ [/tex]
Hence,
[tex]\:\boxed{\begin{aligned}& \:\sf \:(1). \: 9 \: is \: mean \: continued \: proportion \: between \: 3 \: and \: 27 \: \\ \\& \:\sf \: (2). \: 4 \: consective \: odd \: numbers \: are \: 21, \: 23, \: 25, \: 27\end{aligned}} \qquad \: \\ \\ [/tex]