The 6th term of an a.p is 35 and the 13th term is 77.calculate s6-s4
Answer
a+5d = 35
a+12d = 77
find a and d, and then
S6-S4 = 6/2 (2a+5d) - 4/2 (2a+3d)
or, more simply, S6-S4 = a5+a6 = a+4d + a+5d = 2a+9d
To calculate the difference s6 - s4, we first need to find the value of both s6 and s4.
Given information:
a6 = 35 (6th term)
a13 = 77 (13th term)
To find s6, we can use the formula for the sum of an arithmetic progression (AP):
Sn = n/2 * (2a + (n-1)d),
where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
In this case, we need to find s6, so n = 6 and a = a1 (first term). Let's say a1 = a.
Substituting the given values:
35 = (6/2) * (2a + (6-1)d)
35 = 3 * (2a + 5d)
Similarly, for s4, we need to find the sum of the first four terms:
Sn = n/2 * (2a + (n-1)d)
Substituting the values:
S4 = (4/2) * (2a + (4-1)d)
S4 = 2 * (2a + 3d)
Now, our task is to calculate the difference s6 - s4.
s6 - s4 = [3 * (2a + 5d)] - [2 * (2a + 3d)]
= 6a + 15d - 4a - 6d
= 2a + 9d
Now, we need more information about either the common difference (d) or the first term (a) to calculate the difference s6 - s4 more precisely.
To calculate the difference between the 6th term (S6) and the 4th term (S4) of an arithmetic progression (AP), we first need to find the values of S6 and S4.
We are given that the 6th term of the AP is 35. This means that S6 is equal to 35.
To find S4, we can use the formula for the nth term of an AP: Tn = a + (n-1)d, where Tn is the nth term, a is the first term, n is the number of terms, and d is the common difference.
We know that the 1st term (a) is equal to T1, and the common difference (d) is equal to T2 - T1.
Let's calculate T1, T2, and T3 to find the common difference (d):
T1 = a
T2 = a + d
T3 = a + 2d
Now, we can substitute the given values into the equations:
T1 = a
T2 = a + d
T3 = a + 2d
We are given that T6 = 35, so we have:
T6 = a + 5d = 35
Similarly, T13 = 77, so:
T13 = a + 12d = 77
We have a system of two equations with two variables:
a + 5d = 35 ...(1)
a + 12d = 77 ...(2)
To solve this system, we can use the method of substitution or elimination.
Let's use the substitution method. Solving equation (1) for a, we have:
a = 35 - 5d
Substituting this expression for a into equation (2), we get:
35 - 5d + 12d = 77
Combine like terms:
7d = 42
Divide both sides by 7:
d = 6
Now, substitute the value of d back into equation (1) to find a:
a + 5(6) = 35
a + 30 = 35
a = 35 - 30
a = 5
So, the first term (a) is 5 and the common difference (d) is 6.
Now, we can find S4:
S4 = a + 3d (since the 4th term is a + 3d)
Substituting the values of a and d, we have:
S4 = 5 + 3(6) = 5 + 18 = 23
Finally, we can calculate the difference between S6 and S4:
S6 - S4 = 35 - 23 = 12
Therefore, the difference between the 6th term (S6) and the 4th term (S4) of the given arithmetic progression is 12.