In an A.P. having 100 terms. Its first term is 3, M=5n and Sm/Sn does not depend on n then find second term.
What are you doing man......
Just tell me the final answer.....
To find the second term of the arithmetic progression (A.P.), we are given the following information:
- The A.P. has 100 terms.
- The first term (a₁) is 3.
- The formula for the middle term (M) is M = 5n.
- The sum of the first m terms (Sm) divided by the sum of the first n terms (Sn) does not depend on n.
To find the second term (a₂), we need to analyze the given information step by step:
Step 1: Finding the common difference (d)
In an arithmetic progression, the common difference (d) remains the same throughout the progression. We can find it using the formula for the middle term:
M = a₁ + (n-1)d (Formula for finding the middle term)
5n = 3 + (n-1)d (Substituting a₁ = 3 and M = 5n)
5n - 3 = (n-1)d
5n - 3 = nd - d (Expanding the RHS)
5n = nd - d + 3 (Rearranging the equation)
5n - nd + d = 3 (Combining like terms)
d(1-n) = 3 - 5n (Factoring out d)
d = (3 - 5n) / (1-n) (Dividing both sides by (1-n))
d = -2 (Using the given fact that Sm/Sn does not depend on n)
Therefore, the common difference (d) is -2.
Step 2: Finding the sum of the first 100 terms (S100)
The sum of the first n terms in an arithmetic progression is given by the formula:
Sn = (n/2)[2a₁ + (n-1)d] (Formula for the sum of an A.P.)
Substituting the known values:
Sn = 100/2[2(3) + (100-1)(-2)] (a₁ = 3, d = -2, n = 100)
Sn = 50[6 - 198]
Sn = 50(-192)
Sn = -9600
Therefore, the sum of the first 100 terms (S100) is -9600.
Step 3: Using the fact that Sm/Sn does not depend on n
If the ratio Sm/Sn does not depend on n, it means that for any value of n, the ratio remains constant. Let's denote this constant ratio as "k".
Sm/Sn = k
Substituting the values, we get:
Sm/Sn = S100/Sn = k
S100/(-9600) = k
Step 4: Finding the sum of the first 99 terms (S99)
Using the formula for the sum of an arithmetic progression, we can find the sum of the first 99 terms (S99):
S99 = (99/2)[2a₁ + (99-1)d]
Substituting the known values:
S99 = 99/2[2(3) + (99-1)(-2)] (a₁ = 3, d = -2, n = 99)
S99 = 99/2[6 - 196]
S99 = 99(-190)
S99 = -18810
Step 5: Finding the common ratio (k)
Since we know that Sm/Sn = k, we can now solve for k using the known values of S100 and S99:
S100/(-9600) = k
-9600/(-18810) = k
k ≈ 0.511
Step 6: Finding the second term (a₂)
To find the second term (a₂), we can use the formula for the sum of the first n terms:
Sn = (n/2)[2a₁ + (n-1)d]
Substituting known values of Sn = -9600, a₁ = 3, d = -2, and n = 2:
-9600 = (2/2)[2(3) + (2-1)(-2)]
-9600 = 3 + 2(-2)
-9600 = 3 - 4
-9600 = -1
Therefore, the second term (a₂) is -1.
If I read things correctly, then it does not matter how many terms there are.
Sm/Sn is constant for any n and m=5n
S1 = 3
S5 = 5/2(6+4d) = 15+2d
S2 = 3+d
S10 = 10/2 (6+9d) = 30+45d
Since Sm/Sn is constant,
S5/S1 = S10/S2
(15+2d)/3 = (30+45d)/(3+d)
See what you can do with that