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If a, b, c are in GP and $ 4a,5b,4c $ are in AP such that $ a + b + c = 70, $ then value of b is
 $
  1)5 \\
  2)10 \\
  3)15 \\
  4)20 \\
  $

Answer
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504.3k+ views
Hint: Here we will use the concepts of the GP and AP to get the value of the term “b”. GP
(Geometric progression) is the series in which the ratio of the two consecutive terms always remains the same while in AP (Arithmetic Progression), the difference between the two consecutive terms always remains the same.

Complete step-by-step answer:
Given that: a, b, c are in GP
We can write, $ {b^2} = 4ac $ (by definition)
Also, given that $ 4a,5b,4c $ are in AP
By using the concept,
 $ 2(5b) = 4a + 4c $
Simplifying the above expression –
 $ 10b = 4(a + c) $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \dfrac{{10b}}{4} = (a + c) $
Common factors from the numerator and the denominator cancel each other.
 $ \dfrac{{5b}}{2} = (a + c) $ …. (A)
Also given that-
 $ a + b + c = 70 $
Make the required term the subject and move the other term on the opposite side. When you move any term from one side to the opposite side the sign of the term also changes. Positive term becomes negative and vice-versa.
 $ a + c = 70 - b $ …. (B)
From equation (A) and (B)
 $ \dfrac{{5b}}{2} = 70 - b $
Cross multiply the above expression, where the denominator of one side is multiplied with the numerator of the opposite side.
 $ 5b = 2(70 - b) $
Simplify the above expression –
 $
  5b = 140 - 2b \\
  5b + 2b = 140 \\
  7b = 140 \\
  b = \dfrac{{140}}{7} \;
  $
Common factors from the numerator and the denominator cancel each other.
 $ b = 20 $
Hence, from the given multiple choices – the fourth option is the correct answer.
So, the correct answer is “Option 4”.

Note: Always read the given question statements twice and find the correlation between the given and the required terms. Always remember the difference between GP and AP and apply accordingly. Be careful about the sign convention while simplifying the equations.