if 7 times the 7th term of an A.P is equal to 11 times its eleventh term , what will be its 18th term?
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Let's start by understanding the problem. We are given that 7 times the 7th term of an arithmetic progression (A.P.) is equal to 11 times its eleventh term.
Let's assume that the first term of the A.P. is 'a' and the common difference is 'd'.
The formula to find the nth term of an A.P. is:
Tn = a + (n - 1) * d
According to the problem, we know that:
7 * T7 = 11 * T11
Substituting the formulas for the terms, we can rewrite this equation:
7 * (a + (7 - 1) * d) = 11 * (a + (11 - 1) * d)
Simplifying the equation:
7 * (a + 6d) = 11 * (a + 10d)
Expanding the equation:
7a + 42d = 11a + 110d
Subtracting 7a and 42d from both sides of the equation:
0 = 4a + 68d
Dividing both sides of the equation by 4:
0 = a + 17d
This equation tells us that the value of the first term 'a' is equal to the negative 17 times the common difference 'd'.
Now, let's find the 18th term of the A.P. using the formula:
T18 = a + (18 - 1) * d
Substituting the value of 'a':
T18 = -17d + (18 - 1) * d
T18 = -17d + 17d
T18 = 0
Therefore, the 18th term of the A.P. is 0.
To find the 18th term of an arithmetic progression (A.P), we need to determine the common difference (d) first.
In this case, we are given that 7 times the 7th term is equal to 11 times the 11th term. Let's denote the 7th term as a₇ and the 11th term as a₁₁:
7a₇ = 11a₁₁
Now, we can generalize the formula for the n-th term of an A.P:
aₙ = a + (n-1)d
where a is the first term, d is the common difference, and n is the term number.
Since we don't know the first term or the common difference, let's express a₇ and a₁₁ in terms of a and d:
a₇ = a + 6d
a₁₁ = a + 10d
Next, we can substitute the values of a₇ and a₁₁ into the equation we obtained earlier:
7(a + 6d) = 11(a + 10d)
Expanding the equation yields:
7a + 42d = 11a + 110d
Combine like terms:
-4a = 68d
Divide both sides by -4:
a = -17d
Now we know the first term, a, in terms of the common difference, d.
To find the 18th term (a₁₈), we can substitute n = 18 into the n-th term formula:
a₁₈ = (-17d) + (18-1)d
a₁₈ = -17d + 17d
a₁₈ = 0
Therefore, the 18th term of this arithmetic progression is 0.
7(a+6d) = 11(a+10d)
7a+42d = 11a+110d
4a+68d = 0
a+17d = 0
T18 = a+17d = 0