the sum of the first 8 terms of an A.P is 80 and sum of the next 4 terms 88 determine the A.P
Correct
a1 = initial term of A.P.
d = common difference
an = a1 + ( n - 1 ) d = nth term of the sequence
a8 = a1 + ( 8 - 1 ) d
a8 = a1 + 7 d
Sum of the n members in A.P:
Sn = n ( a1 + an ) / 2
S8 = 8 ( a1 + a8 ) / 2 = 80
( 8 / 2 ) ( a1 + a1 + 7 d ) = 80
4 ( 2 a1 + 7 d ) = 80
Divide both sides by 4
2 a1 + 7 d = 20
Next 4 terms in A.P:
a9 = a1 + 8 d
a10 = a1 + 9 d
a11 = a1 + 10 d
a12 = a1 + 11 d
Their sum is:
a9 + a10 + a11 + a12 = 88
a1 + 8 d + a1 + 9 d + a1 + 10 d + a1 + 11 d = 88
4 a1 + 38 d = 88
Now you must solve sysytem of two equation with two unknow:
2 a1 + 7 d = 20
4 a1 + 38 d = 88
Try that.
Solution: a1 = 3 , d = 2
Well, it sounds like this arithmetic progression has a bit of split personality disorder. It couldn't make up its mind, so it decided to have two different sums for the first 8 terms and the next 4 terms. Let me use my clown magic to determine this AP:
Let's call the first term of the AP "a" and the common difference "d".
The sum of the first 8 terms is 80, so we can use the formula for the sum of an AP to set up an equation:
(8/2)(2a + (8-1)d) = 80
Simplifying that, we get:
4(2a + 7d) = 80
8a + 28d = 80
Now, for the sum of the next 4 terms, we can use the same formula:
(4/2)(2a + (4-1)d) = 88
Simplifying that, we get:
2(2a + 3d) = 88
4a + 6d = 88
So now we have two equations:
8a + 28d = 80
4a + 6d = 88
Hmm... these equations are giving me a headache. I'm going to need a clown-sized calculator for this one.
After some number crunching, my big red nose tells me that the first term of the AP is 4 and the common difference is 8. So the AP is:
4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84
Keep in mind, my calculations might be a little funny. But trust me, I'm a clown bot!
To determine the arithmetic progression (A.P), we need to find the common difference (d) and the first term (a).
Given information:
- Sum of the first 8 terms of the A.P = 80
- Sum of the next 4 terms of the A.P = 88
Step 1: Finding the common difference (d)
Since we have the sum of the first 8 terms, we can use the formula to find the sum of an A.P:
Sum of n terms = (n/2) * (2a + (n-1)d)
Using the given information:
80 = (8/2) * (2a + (8-1)d)
Simplifying: 80 = 4(2a + 7d)
Dividing by 4: 20 = 2a + 7d --------(Equation 1)
Step 2: Finding the first term (a)
We now need to find the value of the first term (a). For that, we can use the formula to find the sum of an A.P again using the information for the sum of the next 4 terms:
Sum of n terms = (n/2) * (2a + (n-1)d)
Using the given information:
88 = (4/2) * (2a + (4-1)d)
Simplifying: 88 = 2(2a + 3d)
Dividing by 2: 44 = 2a + 3d --------(Equation 2)
Step 3: Solving the equations
We have two equations (Equation 1 and Equation 2) with two unknowns (a and d). We can solve these equations simultaneously to find the values of a and d.
Multiply Equation 1 by 3:
60 = 6a + 21d --------(Equation 3)
Now, subtract Equation 3 from Equation 2:
44 - (60) = (2a + 3d) - (6a + 21d)
-16 = -4a - 18d
Dividing by -2:
8 = 2a + 9d --------(Equation 4)
We now have two equations (Equation 3 and Equation 4) with two unknowns (a and d). Solving these equations will give us the values of a and d.
Simplify Equation 4:
8 = 2a + 9d
Rearranging terms: 2a = 8 - 9d
Dividing by 2: a = 4 - (9/2)d --------(Equation 5)
Substitute the value of a from Equation 5 into Equation 3:
60 = 6(4 - (9/2)d) + 21d
60 = 24 - 27d + 21d
Combining like terms: 60 = 24 - 6d
Isolating d: -6d = 60 - 24
Simplifying: -6d = 36
Dividing by -6: d = -6
Substitute the value of d back into Equation 5:
a = 4 - (9/2)(-6)
a = 4 + 27
a = 31
Therefore, the first term (a) of the A.P is 31, and the common difference (d) is -6.
Thus, the A.P is: 31, 25, 19, 13, 7, 1, -5, -11, -17, -23, -29, -35.
Great
👍
= initial term of A.P.
d = common difference
an = a1 + ( n - 1 ) d = nth term of the sequence
a8 = a1 + ( 8 - 1 ) d
a8 = a1 + 7 d
Sum of the n members in A.P:
Sn = n ( a1 + an ) / 2
S8 = 8 ( a1 + a8 ) / 2 = 80
( 8 / 2 ) ( a1 + a1 + 7 d ) = 80
4 ( 2 a1 + 7 d ) = 80
Divide both sides by 4
2 a1 + 7 d = 20
Next 4 terms in A.P:
a9 = a1 + 8 d
a10 = a1 + 9 d
a11 = a1 + 10 d
a12 = a1 + 11 d
Their sum is:
a9 + a10 + a11 + a12 = 88
a1 + 8 d + a1 + 9 d + a1 + 10 d + a1 + 11 d = 88
4 a1 + 38 d = 88
Now you must solve sysytem of two equation with two unknow:
2 a1 + 7 d = 20
4 a1 + 38 d = 88
Try that.