Question

What is the second term of this sequence ?

Answer 1

We are given the sum of the first 3 terms (S₃) and the sum of the first 7 terms (S₇) of an AP. We need to find the second term (a₂).
Given: S₃ = 30 and S₇ = 140.

The sum of the first n terms of an AP is given by Sₙ = (n)/(2)[2a + (n-1)d].
An important property for odd n is that the sum Sₙ is n times the middle term. For S₃, the middle term is a₂. So, S₃ = 3 × a₂.

Method 1: Using the middle term property.
For the sum of the first 3 terms (a₁, a₂, a₃), the middle term is a₂.
The sum is given by S₃ = 3 × (middle term) = 3 × a₂.
We are given S₃ = 30.
3 × a₂ = 30 a₂ = (30)/(3) = 10 Method 2: Using simultaneous equations (longer method).
Using Sₙ = (n)/(2)[2a + (n-1)d]:
For S₃ = 30:
30 = (3)/(2)[2a + (3-1)d] = (3)/(2)[2a + 2d] = 3(a+d) a+d = 10 --- (1)
For S₇ = 140:
140 = (7)/(2)[2a + (7-1)d] = (7)/(2)[2a + 6d] = 7(a+3d) a+3d = 20 --- (2)
Subtracting (1) from (2):
(a+3d) - (a+d) = 20 - 10 2d = 10 d = 5 Substitute d=5 into (1):
a + 5 = 10 a = 5 The second term is a₂ = a + d = 5 + 5 = 10.

The second term of the sequence is 10.

Answer 2

Using the same AP as before (S₃=30, S₇=140), we need to find the fourth term (a₄).

Similar to the previous part, for S₇, the middle term is a₄. So, we can use the property S₇ = 7 × a₄.
Alternatively, we can use the values of a and d found previously (a=5, d=5) and the formula aₙ = a + (n-1)d.

Method 1: Using the middle term property.
For the sum of the first 7 terms (a₁, ..., a₇), the middle term is the ((7+1)/(2)) = 4-th term, which is a₄.
The sum is given by S₇ = 7 × (middle term) = 7 × a₄.
We are given S₇ = 140.
7 × a₄ = 140 a₄ = (140)/(7) = 20 Method 2: Using a and d.
From the previous part, we found a=5 and d=5.
We need to find the fourth term, a₄.
a₄ = a + (4-1)d a₄ = 5 + 3(5) a₄ = 5 + 15 = 20 The fourth term of the sequence is 20.