Question:

The product of three numbers in an $A.P$. is $224$, and the largest number is $7$ times the smallest. Find the numbers.

Updated On: Jun 23, 2023
  • $1, 5, 7 $
  • $2, 6, 8 $
  • $2, 8, 14$
  • $4, 8, 10$
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The Correct Option is C

Solution and Explanation

Let the three numbers be $a - d, a, a + d (d > 0)$ Now $(a- d) \,a(a + d) = 224$ $ \Rightarrow a(a^2 - d^2) = 224 \quad . . .(i)$ Also, $a + d = 7 (a- d)$ $ \Rightarrow d = \frac{3a}{4}$ Substituting this value of $d$ in $(i)$, we get $a\left(a^{2} - \frac{9a^{2}}{16}\right) = 224$ $\Rightarrow a = 8 $ and $d= \frac{3}{4} \times 8 = 6$ Hence, the three numbers are $2, 8,14$.
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Notes on Arithmetic Progression

Concepts Used:

Arithmetic Progression

Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.

For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.

In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.

For eg:- 4,6,8,10,12,14,16

We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.

Read More: Sum of First N Terms of an AP