The sums of the first 8 terms and first 20 terms of an AP are 56 and 260 respectively; find the first term (a) and the common difference (d).
To find the first term (a) and the common difference (d) of an arithmetic progression (AP), we can use the formulas for the sum of the first n terms of an AP:
Sn = (n/2) * (2a + (n-1)d)
Given that the sum of the first 8 terms (S8) is 56, we can use this formula to get the equation:
56 = (8/2) * (2a + (8-1)d)
Simplifying the equation:
56 = 4a + 28d ----- (1)
Similarly, given that the sum of the first 20 terms (S20) is 260, we can use the same formula to get another equation:
260 = (20/2) * (2a + (20-1)d)
Simplifying the equation:
260 = 10a + 190d ----- (2)
Now we have a system of two equations (equation (1) and equation (2)) with two unknowns (a and d). We can solve this system of equations to find the values of a and d.
To solve this system of equations, we can use any algebraic method such as substitution or elimination. Let's use the method of elimination:
Multiply equation (1) by 10:
560 = 40a + 280d ----- (3)
Now subtract equation (3) from equation (2):
260 - 560 = (10a + 190d) - (40a + 280d)
-300 = -30a - 90d
Divide both sides by -30:
10 = a + 3d ----- (4)
We now have a new equation (equation (4)) with a and d as unknowns. Using this equation, we can eliminate one variable to solve for the other. Let's solve for a in terms of d:
a = 10 - 3d ----- (5)
Substitute equation (5) into equation (1):
56 = 4(10-3d) + 28d
56 = 40 - 12d + 28d
56 = 40 + 16d
16d = 56 - 40
16d = 16
d = 1
Substitute the value of d into equation (5) to find the value of a:
a = 10 - 3(1)
a = 10 - 3
a = 7
Therefore, the first term (a) of the AP is 7 and the common difference (d) is 1.
8/2 (2a+7d) = 56
20/2 (2a+19d) = 260
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