The 9th term of an A.P is three times the 5th term. Is it true or false that the 8th term is five times the 4th term.
To determine whether the statement is true or false, we can use the formula for the nth term of an arithmetic progression (A.P):
nth term = a + (n - 1)d
where a is the first term, n is the position term, and d is the common difference.
Let's assume that the first term is a and the common difference is d.
Given that the 9th term (a + 8d) is three times the 5th term (a + 4d), we can write the equation as:
a + 8d = 3(a + 4d)
Expanding the equation:
a + 8d = 3a + 12d
Rearranging the terms:
8d - 12d = 3a - a
Simplifying, we get:
-4d = 2a
From this equation, we can conclude that for the 9th term to be three times the 5th term, the common difference (d) must be equal to half of the first term (a).
Now, let's check if the 8th term (a + 7d) is indeed five times the 4th term (a + 3d).
Assuming we have the required conditions, which are d = a/2, we can substitute this value into the equation:
a + 7(a/2) = 5(a + 3(a/2))
Expanding the equation:
a + (7/2)a = 5a + 15(a/2)
Rearranging the terms:
a + (7/2)a - 15(a/2) = 5a
Simplifying, we get:
a/2 = 5a
This equation is not possible since it states that one-half of the first term is equal to five times the first term. This contradicts the assumption that d = a/2.
Therefore, it is false to say that the 8th term is five times the 4th term.
To determine whether the statement is true or false, we must first find the values of both the 9th and 8th terms in the arithmetic progression (A.P).
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is the same.
Let's consider the 9th term, denoted as a₉, and the 5th term, denoted as a₅. According to the problem statement, a₉ = 3 * a₅.
Now, we need to determine the value of the 8th term, denoted as a₈, and the 4th term, denoted as a₄.
Since it is not provided in the question, we do not know the common difference (d), which is the difference between each consecutive term. Therefore, we have to express a₈ and a₄ in terms of a₅.
The general formula for the nth term (aₙ) of an A.P is given as:
aₙ = a₁ + (n-1)d
Using this formula, we can express a₈ and a₄ in terms of a₅:
a₈ = a₁ + 7d
a₄ = a₁ + 3d
Now, we can proceed to solve the problem.
Since a₉ = 3 * a₅, we can express a₉ in terms of a₅:
a₉ = a₅ + 4d
Comparing the expressions for a₈ and a₄ with a₉, we observe that both are expressed using the same common difference (d).
If a₉ = a₅ + 4d, and a₅ = 3 * a₅, we can substitute a₅ + 4d into a₉:
3 * a₅ = a₅ + 4d
2 * a₅ = 4d
a₅ = 2d
Therefore, we have expressed a₅ in terms of the common difference (d).
Now, we can substitute a₅ = 2d into the expressions for a₈ and a₄:
a₈ = a₁ + 7d
a₈ = a₁ + 7(2d)
a₈ = a₁ + 14d
a₄ = a₁ + 3d
a₄ = a₁ + 3(2d)
a₄ = a₁ + 6d
So, the statement "the 8th term is five times the 4th term" is equivalent to: a₈ = 5 * a₄.
Substituting the expressions for a₈ and a₄ obtained earlier:
a₁ + 14d = 5 * (a₁ + 6d)
a₁ + 14d = 5a₁ + 30d
-4a₁ = 16d
a₁ = -4d
Since we have expressed a₁ in terms of the common difference (d), we can conclude that the statement "the 8th term is five times the 4th term" is TRUE.
a+8d = 3(a+4d)
a+8d = 3a+12d
-2a = 4d
a = -2d
so, is
a+7d = 5(a+3d) ?
Substitute from the first equation and find out.