The sum of the first four terms of a linear sequence is 26,that of the next four terms is 74, find the values of the first term and the common difference
just use the formulas you know.
4/2 (2a+3d) = 26
Now, the next four terms' sum is the sum of the first 8 terms, minus the sum above
8/2 (2a+7d) - 26 = 74
Now just solve for a and d
a + a+d + a+2d + a+ 3d = 26
a+4d + a+5d + a+6d + a+7d = 74
well
4 a + 6 d = 26
4 a + 22 d = 74
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0 - 16 d = - 48
d = 3
etc
To find the values of the first term and common difference in a linear sequence, we can use the formula for the sum of an arithmetic series.
The sum of the terms of an arithmetic (linear) sequence can be calculated using the formula:
S = (n/2)(2a + (n-1)d)
Where:
- S represents the sum of the terms
- n represents the number of terms
- a represents the first term
- d represents the common difference
In this case, we know the sum of the first four terms is 26, so we can substitute these values into the formula:
26 = (4/2)(2a + (4-1)d)
Simplifying, we get:
26 = 2(2a + 3d)
13 = 2a + 3d ---(equation 1)
Similarly, we can calculate the sum of the next four terms, which is 74:
74 = (4/2)(2a + (8-1)d)
74 = 2(2a + 7d)
37 = 2a + 7d ---(equation 2)
We now have a system of two equations (equation 1 and equation 2) with two unknowns (a and d). We can solve this system using a method like substitution or elimination.
Let's use elimination to find the values of a and d:
Multiply equation 1 by 7 and equation 2 by 3 to make the coefficients of 'd' the same:
91 = 14a + 21d
111 = 6a + 21d
If we subtract the second equation from the first equation, we can eliminate 'd':
91 - 111 = 14a - 6a
-20 = 8a
Divide both sides by 8:
a = -20/8
a = -2.5
Now substitute the value of a back into equation 1 or equation 2 to calculate the value of 'd'. Let's use equation 1:
13 = 2(-2.5) + 3d
13 = -5 + 3d
3d = 13 + 5
3d = 18
d = 18/3
d = 6
Therefore, the first term (a) is -2.5 and the common difference (d) is 6.