A sequence of numbers U1, U2, U3 ..., Un satisfies the relation Un+1 = Un + (1/2)^2 for all integers n >_1 If U1 = 1, find: the values of U2, U3 and U4 an expression for Un in terms of n. the value of Un for large values of n
To solve that:
Un+1 = Un¬+(1/2)2
When n =1
U2=1+1/4
U2=5/4
When n=2
U3=5/4 + 1/4
U3=6/4
When n=3
U4=6/4+1/4
U4=7/4
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nwoforchibuikem
To find the values of U2, U3, and U4, we can use the given relation Un+1 = Un + (1/2)^2.
Starting with U1 = 1, we can find U2 as follows:
U2 = U1 + (1/2)^2
= 1 + (1/4)
= 1.25
To find U3, we substitute U2 back into the relation:
U3 = U2 + (1/2)^2
= 1.25 + (1/4)
= 1.50
Next, to find U4, we substitute U3 back into the relation:
U4 = U3 + (1/2)^2
= 1.50 + (1/4)
= 1.75
Now, let's find an expression for Un in terms of n.
We already know the values of U1, U2, U3, and U4, so let's observe a pattern based on these values:
U1 = 1
U2 = 1 + (1/4) = 1.25
U3 = 1.25 + (1/4) = 1.50
U4 = 1.50 + (1/4) = 1.75
From this pattern, we can deduce that Un = 1 + (n-1) * (1/4) = (4n-3) / 4.
Now, let's calculate the value of Un for large values of n.
As n approaches infinity, the term (4n-3) in the expression for Un dominates, and the (1/4) term becomes insignificant. Therefore, for large values of n, we can approximate Un as:
Un ≈ (4n-3) / 4
This approximation will work well for very large values of n.
arithmetic sequence ?
a = 1
d = 1/2 * 1/2 = 1/4
Tn = a + (n-1) d
Un = 1 + (n-1)* 1/4