Find the nth term and first three terms of the AP having U4=93and U8= -65
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.
To find the nth term of an arithmetic progression (AP), we need to determine the common difference (d) first. The formula to calculate the nth term of an AP is:
Un = U1 + (n-1)d
Given U4 = 93 and U8 = -65, we can form two equations:
U4 = U1 + 3d ...(1)
U8 = U1 + 7d ...(2)
Now, we can solve these equations to find the values of U1 and d.
Subtracting equation (1) from equation (2) to eliminate U1:
U8 - U4 = 7d - 3d
-65 - 93 = 4d
-158 = 4d
d = -158/4
d = -39.5
Substituting the value of d in equation (1) to find U1:
U4 = U1 + 3d
93 = U1 + 3(-39.5)
93 = U1 - 118.5
U1 = 93 + 118.5
U1 = 211.5
So, the common difference (d) is -39.5 and the first term (U1) is 211.5.
Now, we can find the nth term by using the formula:
Un = U1 + (n-1)d
Substituting the values of U1 and d:
Un = 211.5 + (n-1)(-39.5)
Un = 211.5 - 39.5n + 39.5
Un = 251 + (-39.5n)
Therefore, the nth term of the AP is -39.5n + 251.
To find the first three terms, we substitute n = 1, 2, and 3 into the nth term formula:
U1 = -39.5(1) + 251 = 211.5
U2 = -39.5(2) + 251 = 172.5
U3 = -39.5(3) + 251 = 133.5
So, the first three terms of the AP are 211.5, 172.5, and 133.5.
To find the nth term and the first three terms of an arithmetic progression (AP), we need two pieces of information - two terms of the AP.
Given the information that U4 = 93 and U8 = -65, we can calculate the common difference (d) and use it to find the nth term and the first three terms.
Step 1: Calculate the common difference (d)
The common difference (d) is the difference between consecutive terms in an AP. Subtracting any two terms will give us the common difference.
d = U8 - U4
d = (-65) - (93)
d = -158
Step 2: Find the first term (a)
To find the first term (a) of the AP, we can use the formula:
a = U4 - (4-1) * d
a = 93 - 3 * (-158)
a = 93 + 474
a = 567
Step 3: Find the nth term (Un)
The formula to find the nth term (Un) of an AP is:
Un = a + (n-1) * d
Substituting the values we have,
Un = 567 + (n-1) * (-158)
Un = 567 - 158n + 158
Un = 725 - 158n
Step 4: Find the first three terms
To find the first three terms of the AP, substitute the values of n = 1, 2, and 3 into the nth term formula:
U1 = 725 - 158(1) = 567
U2 = 725 - 158(2) = 409
U3 = 725 - 158(3) = 251
Therefore, the nth term (Un) of the given AP is 725 - 158n, and the first three terms are 567, 409, and 251.