Question

Find the nth term and first three terms of the AP having U4=93and U8= -65

Answer 1

To find the nth term of an arithmetic progression (AP), we need to find the common difference (d).

Given that U4 = 93 and U8 = -65, we can find the common difference using the formula:

Uₙ = U₁ + (n - 1)d

For U4 = 93:
93 = U₁ + (4 - 1)d
93 = U₁ + 3d ---> Equation 1

For U8 = -65:
-65 = U₁ + (8 - 1)d
-65 = U₁ + 7d ---> Equation 2

We now have two equations with two variables (U₁ and d). We can solve this system of equations to find the values of U₁ and d.

Subtracting Equation 2 from Equation 1, we get:
93 - (-65) = U₁ + 3d - U₁ - 7d
158 = -4d

Dividing both sides by -4, we find:
d = -39.5

Substituting the value of d in Equation 1, we can solve for U₁:
93 = U₁ + 3(-39.5)
93 + 118.5 = U₁
U₁ = 211.5

Therefore, the first term (U₁) of the AP is 211.5, and the common difference (d) is -39.5.

To find the nth term of the AP, we can now use the formula:

Uₙ = U₁ + (n - 1)d

Let's find the 2nd, 3rd, and 4th terms of the AP.

For n = 2:
U₂ = 211.5 + (2 - 1)(-39.5)
U₂ = 211.5 - 39.5
U₂ = 172

For n = 3:
U₃ = 211.5 + (3 - 1)(-39.5)
U₃ = 211.5 - 79
U₃ = 132.5

For n = 4:
U₄ = 211.5 + (4 - 1)(-39.5)
U₄ = 211.5 - 118.5
U₄ = 93

The nth term of the AP is Uₙ = 211.5 - 39.5(n - 1) or 250 - 39.5n.

The first three terms of the AP are: 211.5, 172, and 132.5.