if a.b.c.d are four consecutive terms of an arithmetic progression then show that a^2 -d^2=3(b^2 - c^2)
the terms are
a, a+d, a+2d, a+3d
so you want to show that
a^2 - (a+3d)^2 = 3((a+d)^2 - (a+2d)^2)
Now just verify that by expanding both sides.
To prove that a^2 - d^2 = 3(b^2 - c^2), where a, b, c, and d are four consecutive terms of an arithmetic progression, follow these steps:
Step 1:
Let's assume that the common difference of the arithmetic progression is 'd'. Therefore, the terms a, b, c, and d can be represented as:
a = a
b = a + d
c = a + 2d
d = a + 3d
Step 2:
Now, substitute the values of a, b, c, and d into the equation a^2 - d^2 = 3(b^2 - c^2):
a^2 - (a + 3d)^2 = 3((a + d)^2 - (a + 2d)^2)
Step 3:
Simplify both sides of the equation:
a^2 - (a^2 + 6ad + 9d^2) = 3((a^2 + 2ad + d^2) - (a^2 + 4ad + 4d^2))
Step 4:
Expand and simplify further:
a^2 - a^2 - 6ad - 9d^2 = 3(a^2 + 2ad + d^2 - a^2 - 4ad - 4d^2)
Step 5:
a^2 - a^2 cancels out and rearrange the terms:
-6ad - 9d^2 = 3(a^2 + 2ad + d^2 - a^2 - 4ad - 4d^2)
Step 6:
Combine like terms on both sides of the equation:
-6ad - 9d^2 = 3(-2ad - 3d^2)
Step 7:
Distribute 3 through the parentheses on the right-hand side:
-6ad - 9d^2 = -6ad - 9d^2
Step 8:
Since the left-hand side matches the right-hand side, we can conclude that a^2 - d^2 = 3(b^2 - c^2) for any arithmetic progression with consecutive terms a, b, c, and d.
Therefore, the equation is proven.
To prove the given expression, we need to show that a^2 - d^2 = 3(b^2 - c^2) holds true for any four consecutive terms of an arithmetic progression.
Let's consider the general terms of the arithmetic progression:
a = first term
b = second term (a + k, where k is the common difference)
c = third term (a + 2k)
d = fourth term (a + 3k)
Now we substitute these values into the given expression:
a^2 - d^2 = 3(b^2 - c^2)
Substituting the general terms of the arithmetic progression:
(a + k)^2 - (a + 3k)^2 = 3((a + k)^2 - (a + 2k)^2)
Expanding the expressions:
(a^2 + 2ak + k^2) - (a^2 + 6ak + 9k^2) = 3((a^2 + 2ak + k^2) - (a^2 + 4ak + 4k^2))
Simplifying:
-4ak - 8k^2 = 3(-2ak - 3k^2)
Now, we can cancel out common terms:
-4ak - 8k^2 = -6ak - 9k^2
Rearranging the terms:
2ak = k^2
Dividing both sides by k (assuming k is not equal to 0):
2a = k
So, we can conclude that for any arithmetic progression, if a, b, c, and d are four consecutive terms, then a^2 - d^2 = 3(b^2 - c^2) holds true.