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Chapter 8: Problem 595

Find \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) between 2 and 18 such that \(\mathrm{a}+\mathrm{b}+\mathrm{c}=25\), \(2, a, b\) are in A. P. and \(b, c, 18\) are in G. P. (A) \(5,8,12\) (B) \(4,8,13\) (C) \(3,9,13\) (D) \(5,9,11\)

Short Answer

Expert verified
The correct answer is (A) $a=5, b=8,$ and $c=12$.

Step by step solution

01

Formulate the Necessary Equations

Since 2, a, b are in A.P, we know that: a - 2 = b - a We can rewrite this as: 2a - b = 2 ---- (1) Similarly, since b, c, 18 are in G.P, we get: c/b = 18/c We can rewrite this as: c² = 18b ---- (2) Additionally, we are given the condition that a+b+c = 25, which we will use later.
02

Solve for Variable 'b'

To solve for variable 'b', multiply equation (1) by 18: 36a - 18b = 36 ---- (3) Now, substitute equation (2) into equation (3): 36a - c² = 36 Recall the given condition a+b+c = 25, which can be rewritten as: a+b = 25 - c Solve for c: c = 25 - a - b Substitute this back into the equation 36a - c² = 36: 36a - (25-a-b)² = 36 ---- (4)
03

Solve for Variables 'a' and 'c'

Now, we can substitute a = (b+2) from equation (1) into equation (4): 36(b+2) - (25-(b+2)-b)² = 36 Expand and simplify to get: 13b - b² = 8 ---- (5) We can try each option to see which one satisfies equation (5) for a, b, and c: (A) a=5, b=8, c=12 (B) a=4, b=8, c=13 (C) a=3, b=9, c=13 (D) a=5, b=9, c=11 Let's check each option: (A) 13(8) - (8)² = 8 (True) (B) 13(8) - (4)² = 8 (False) (C) 13(9) - (3)² = 8 (False) (D) 13(9) - (5)² = 8 (False) Only option (A) satisfies the equation, so the answer is: a = 5, b = 8, c = 12 (A)

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Most popular questions from this chapter

\([1 /(2 \times 5)]+[1 /(5 \times 8)]+[1 /(8 \times 11)]+\ldots 100\) terms (A) \([(25) /(160)]\) (B) \((1 / 6)\) (C) \([(25) /(151)]\) (D) \([(25) /(152)]\)

If the A. M. of two numbers a and \(b\) is equal to \(\sqrt{(10) \text { times }}\) their G. M. then \([(a-b) /(a+b)]=\) (A) \([\sqrt{(10) / 3]}\) (B) \(3 \sqrt{(10)}\) (C) \([9 /(10)]\) (D) \([3 / \sqrt{(10)}]\)

If \([(b+c-a) / a],[(c+a-b) / b],[(a+b-c) / c]\) are in A. \(P\) then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in (A) G P. (B) A. P. (C) H. P. (D) A. G P.

If \([1 /(\mathrm{b}-\mathrm{c})],[1 /(2 \mathrm{~b}-\mathrm{x})]\) and \([1 /(\mathrm{b}-\mathrm{a})]\) are in A. P., then \(\mathrm{a}-(\mathrm{x} / 2), \mathrm{b}-(\mathrm{x} / 2), \mathrm{c}-(\mathrm{x} / 2)\) are in (A) A. P. (B) G. P. (C) H. P. (D) A. G P.

If \(1, \mathrm{~A}_{1}, \mathrm{~A}_{2}, \mathrm{~A}_{3}, \ldots \mathrm{A}_{\mathrm{n}}, 31\) are in A. P. and \(\mathrm{A}_{7}: \mathrm{A}_{\mathrm{n}-1}=5: 9\) then \(\mathrm{n}=\) (A) 28 (B) 14 (C) 15 (D) 13
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