In an arithmetic series the 3rd term is twice the 8th term. Find the sum of the first 25 terms.
Tn = a + a+d + a+2d .... + a +(n-1) d
T3 = a +2d
T8 = a + 7 d
so
a + 2d = 2 (a+7d)
a + 2d = 2 a + 14 d
a + 12 d = 0
a = -12 d
sum of first 25 = (n/2)[ 2a +(n-1)d ]
= (25/2) [ -24 d + (24) d ]
LOL !!!
= 0
oo hanks i am doing this sum form hours to hours .ur great
To find the sum of the first 25 terms of an arithmetic series, we need to determine the first term (a) and the common difference (d).
Let's start by assigning variables to the given information.
Given:
The 3rd term (a3) is twice the 8th term (a8).
We can express this information mathematically:
a3 = 2 * a8
Now let's use the formula for the nth term of an arithmetic sequence:
an = a + (n-1)d
We know that a3 = a + 2d and a8 = a + 7d. Substituting these values into the given equation, we get:
a + 2d = 2(a + 7d)
Now we can solve this equation to find the values of a and d.
Expanding the equation:
a + 2d = 2a + 14d
Simplifying:
-a = 12d
Dividing both sides by -12:
a = -1/12d
So we have found the relationship between the first term a and the common difference d in the arithmetic series.
Now let's find the value of d.
We can use the formula for the nth term of an arithmetic sequence:
an = a + (n-1)d
Since the 8th term is a8 = a + 7d, we can substitute -1/12d for a to get:
a8 = -1/12d + 7d
To further simplify, we can find a common denominator:
a8 = -1/12d + 7d * (12/12)
a8 = -1/12d + 84d/12
a8 = (84d - 1)/12
Now we know that the 8th term of the arithmetic series is (84d - 1)/12.
Next, let's find the value of a3 by substituting the values we have found:
a3 = -1/12d + 2d
We can combine the terms by finding a common denominator:
a3 = -1/12d + 24d/12
a3 = (24d - 1)/12
From the given equation, we know that a3 = 2 * a8, so:
(24d - 1)/12 = 2 * (84d - 1)/12
Expanding the equation:
24d - 1 = 2 * (84d - 1)
Simplifying:
24d - 1 = 168d - 2
Rearranging the equation:
168d - 24d = 2 - 1
144d = 1
d = 1/144
Now that we know the value of d, we can find the value of a:
a = -1/12d
a = -1/12 * 1/144
a = -1/1728
So the first term (a) is -1/1728 and the common difference (d) is 1/144.
Now we can find the sum of the first 25 terms using the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Substituting the values we have found:
S25 = (25/2)(2 * (-1/1728) + (25-1) * (1/144))
Calculating the expression inside the brackets:
(-1/864) + (24/144) = (-1/864) + (1/6) = (1 - 144) / 864 = -143/864
Substituting this value back into the formula:
S25 = (25/2)(-143/864)
Calculating the product:
S25 = (25 * (-143)) / (2 * 864)
Simplifying:
S25 = (-3575) / 3456
Therefore, the sum of the first 25 terms of the arithmetic series is approximately -1.0359.