Advertisements
Advertisements
प्रश्न
If a, b, c, d are in continued proportion, prove that (b − c)2 + (c − a)2 + (d − b)2 = (d − a)2.
Advertisements
उत्तर
Since a, b, c, d are in continued proportion, we have
`a/b = b/c = c/d` = K (say)
∴ c = dK, b = cK = dK2 and a = bK = dK3.
L.H.S.
= (b − c)2 + (c − a)2 + (d − b)2
= (dK2 − dK)2 + (dK − dK3)2 + (d − dK2)2
= d2K2(K − 1)2 + d2K2 (1 − K2)2 + d2 (1 − K2)2
= d2 [K2(K − 1)2 + K2(K2 − 1)2 + d2(k2 − 1)2]
= d2 [K2 (K − 1)2 + K2 (K − 1)2 (K + 1)2 + (K − 1)2 (K + 1)2]
= d2 (K − 1)2 [K2 + K2 (K+ 1)2 + (K + 1)2]
= d2 (K − 1)2 [K2 + K2 (K2 + 2K + 1) + K2 + 2K + 1]
= d2 (k − 1)2 [K4 + 2K3 + 3K2 + 2K + 1]
= d2(K − 1)2 (K2 + K + 1)2
= d2[(K − 1) (K2 + K + 1)2]
= d2 (K3 − 1)2 = (dK3 − d)2 = (a − d)2 = (d − a)2 = R.H.S.
Hence, (b − c)2 + (c − a)2 + (d − b)2 = (d − a)2.
संबंधित प्रश्न
What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
If a, b, c are in continued proportion and a(b – c) = 2b, prove that: `a - c = (2(a + b))/a`.
Verify the following:
91 : 104 : : 119 : 136
Determine if the following ratio form a proportion:
25 cm : 1 m and Rs 40 : Rs 160
In proportion, the 1st, 2nd, and 4th terms are 51, 68, and 108 respectively. Find the 3rd term.
If 15 tins of the same size contain 234 kg of oil, how much oil will there be in 10 such tins?
Are 30, 40, 45, and 60 in proportion?
10 books is to 15 books as 3 books is to 15 books
Bachhu Manjhi earns Rs. 24000 in 8 months. At this rate, how much does he earn in one year?
Determine if the following are in proportion.
15, 45, 40, 120
