the first and second terms of an arithmetic progression are 161 and 154 respectively.the sumthe first m terms is zero.find the value of m
The answer will be 47.(329/7=47).
Why did the mathematician join the circus? Because he wanted to find the value of "m" surrounded by clowns like me!
Now, let's solve this puzzle. In an arithmetic progression, the difference between consecutive terms is constant. We can find the common difference using the formula:
Common difference (d) = (Second term - First term) = 154 - 161 = -7
Now, let's find the value of "m" using the formula for the sum of an arithmetic progression:
Sum (S) = (m/2)(2a + (m-1)d)
Since the sum of the first "m" terms is zero, we have:
0 = (m/2)(2(161) + (m-1)(-7))
Now, let's simplify this equation:
0 = (m/2)(322 - 7m + 7)
0 = (m/2)(329 - 7m)
Since we know that at least one term must be non-zero to have a sum greater than zero, we can divide both sides of the equation by (m/2):
0 = (329 - 7m)
Rearranging the equation gives us:
7m = 329
Finally, solving for "m":
m = 329 / 7
m ≈ 47
So, the value of "m" is approximately 47. Keep in mind that this answer assumes that there is no mistake in the information provided.
a = 161
a+d = 154 , so d = -7
S(m) = (m/2)(2a + (m-1)d)
0 = m/2(322 - 7m+ 7)
329m - 7m^2=0
49m - m^2 = 0
m(49-m)=0
m = 49
To find the value of m, we need to determine the common difference of the arithmetic progression first.
The common difference (d) can be found using the formula:
d = (second term - first term)
Using the given information:
d = (154 - 161)
d = -7
Now that we know the common difference, we can find the value of m by using the formula for the sum of an arithmetic progression:
Sᵤ = (u/2) * (2a + (u-1)d)
Since the sum of the first m terms is zero (Sᵤ = 0), we can set up the equation:
0 = (m/2) * (2 * 161 + (m-1) * -7)
Simplifying the equation further:
0 = (m/2) * (322 - 7m + 7)
0 = (m/2) * (329 - 7m)
Now we can solve for m:
(m/2) * (329 - 7m) = 0
Since the product of two factors is zero, one of the factors must be zero:
m/2 = 0 or 329 - 7m = 0
Solving the equations:
m = 0 * 2 or -7m = -329
m = 0 or m = -329/-7
m = 0 or m ≈ 47
Considering that the number of terms (m) cannot be negative, the value of m is 47.
To find the value of "m," we need to determine the number of terms in the arithmetic progression such that the sum of the first "m" terms is zero.
In an arithmetic progression, the nth term (Tn) can be determined using the formula:
Tn = a + (n - 1)d
Where:
a = first term
d = common difference between the terms
n = number of terms
Given that the first term (a) is 161 and the second term is 154, we can find the common difference (d):
d = second term - first term
d = 154 - 161
d = -7
Now let's find the formula for the sum of an arithmetic progression:
Sn = (n/2)(2a + (n - 1)d)
Given that the sum of the first "m" terms is zero, we can set up the equation:
0 = (m/2)(2a + (m - 1)d)
Plugging in the known values:
0 = (m/2)(2*161 + (m - 1)*(-7))
Simplifying further:
0 = m(322 - 7(m - 1))
0 = m(322 - 7m + 7)
0 = m(329 - 7m)
To find the value of "m," we can consider two cases:
Case 1: m = 0
In this case, the sum of the first "0" terms would indeed be zero.
Case 2: 329 - 7m = 0
Solving this equation, we get:
329 = 7m
m = 329/7
m ≈ 47
Therefore, the value of "m" is either 0 or approximately 47, depending on how we interpret the problem.