The 8th term of an ap is 5times the 3rd while the 7th term is 9 greater than the 4th term.write the first 5 terms.
a+7d = 5(a+2d)
a+6d = a+3d+9
solve for a and d, and then you can write the sequence.
Bangalore
I don't no it
To find the first five terms of an arithmetic progression (AP), we need to have either the common difference (d) or enough information to calculate it.
Let's start by using the given information:
- The 8th term (𝑛 = 8) is 5 times the 3rd term (𝑛 = 3).
- The 7th term (𝑛 = 7) is 9 greater than the 4th term (𝑛 = 4).
From the information given, we can set up two equations to solve for the common difference:
Equation 1: 𝑎₈ = 5𝑎₃
Equation 2: 𝑎₇ = 𝑎₄ + 9
To find 𝑎₇ and 𝑎₈, we first need to find 𝑎₄ and 𝑎₃. To do this, we must find the common difference (𝑑).
Using Equation 2, we can rearrange it to solve for 𝑎₄:
𝑎₇ = 𝑎₄ + 9
𝑎₄ = 𝑎₇ - 9
Now, using Equation 1, we can solve for the common difference (𝑑):
𝑎₈ = 5𝑎₃
𝑎₈ = 𝑎₄ + 3𝑑 ⟶ substituting 𝑎₄ = 𝑎₇ - 9
𝑎₈ = (𝑎₇ - 9) + 3𝑑
𝑎₈ = 𝑎₇ + 3𝑑 - 9
Since 𝑎₈ is 5 times 𝑎₃, this gives us an equation:
𝑎₇ + 3𝑑 - 9 = 5𝑎₃
Now we have two equations with two unknowns, 𝑎₃ and 𝑑:
1. 𝑎₇ - 9 = 𝑎₄ ⟶ Equation 2 rearranged
2. 𝑎₇ + 3𝑑 - 9 = 5𝑎₃ ⟶ Equation 1 rearranged
To simplify, we combine the two equations:
(𝑎₇ - 9) + 3𝑑 - 9 = 5(𝑎₇ - 9)
𝑎₇ - 18 + 3𝑑 = 5𝑎₇ - 45
3𝑑 - 5𝑎₇ + 27 = 0
3𝑑 - 5(𝑎₇ + 3) = 0
3𝑑 - 5𝑎₇ - 15 = 0
This equation allows us to find the value of the common difference (𝑑) in terms of 𝑎₇:
𝑑 = (5𝑎₇ + 15)/3 ⟶ Equation 3
Now, substituting 𝑎₄ = 𝑎₇ - 9 into Equation 1:
𝑎₈ = 𝑎₇ + 3𝑑 - 9
𝑎₈ = 𝑎₇ + 3[(5𝑎₇ + 15)/3] - 9 ⟶ substituting 𝑑 from Equation 3 into Equation 1
𝑎₈ = 𝑎₇ + 5𝑎₇ + 15 - 9
𝑎₈ = 6𝑎₇ + 6
By substituting values of 𝑎₇ into 𝑎₈ = 6𝑎₇ + 6, we can find the 8th term.
To find the first 5 terms, we need to plug in 𝑛 = 1, 2, 3, 4, and 5 into the formula 𝑎𝑛 = 𝑎₁ + (𝑛−1)𝑑, where 𝑎₁ is the first term and 𝑑 is the common difference.
Let's start with 𝑛 = 1:
𝑎₁ = 𝑎₁ + (1-1)𝑑 = 𝑎₁
The formula shows that the first term (𝑎₁) is itself. Since we don't have any information about 𝑎₁, we cannot find its value.
Next, let's find 𝑎₂:
𝑎₂ = 𝑎₁ + (2-1)𝑑 = 𝑎₁ + 𝑑
Since we don't know 𝑎₁ and 𝑑, we cannot find 𝑎₂.
Now let's find 𝑎₃:
𝑎₃ = 𝑎₁ + (3-1)𝑑 = 𝑎₁ + 2𝑑
Still, we don't have enough information to solve for 𝑎₃ since we don't know 𝑎₁ and 𝑑.
Similar calculations can be made for 𝑎₄ and 𝑎₅, but we still don't have enough information to calculate them.
Therefore, without more information, we cannot determine the first five terms of the arithmetic progression (AP).