consider the sequence given by U1=-0.3 Un+1=Un+0.7 i)state what type of sequence this is. ii)write down the first 4 terms of the sequence. iii) find a closed form for sequnce. iv) use the closed form to find the nth term of the sequnce when Un=36.1 (n=1,2,3 ETC)
i) To determine the type of sequence, we need to observe the pattern of the terms. In this case, each term is obtained by adding a constant value of 0.7 to the previous term. This indicates that the sequence is an arithmetic sequence.
ii) Writing down the first four terms of the sequence:
U1 = -0.3 (given)
U2 = U1 + 0.7 = -0.3 + 0.7 = 0.4
U3 = U2 + 0.7 = 0.4 + 0.7 = 1.1
U4 = U3 + 0.7 = 1.1 + 0.7 = 1.8
So, the first four terms are:
U1 = -0.3
U2 = 0.4
U3 = 1.1
U4 = 1.8
iii) To find a closed form expression for the sequence, we can observe that each term can be obtained by multiplying the term index (n) by a common difference of 0.7 and subtracting 1 from the result.
Therefore, the closed form expression for the sequence is:
Un = (0.7n - 1)
iv) To find the nth term of the sequence when Un = 36.1, we can substitute Un = 36.1 into the closed form expression and solve for n.
36.1 = 0.7n - 1
0.7n = 36.1 + 1
0.7n = 37.1
Dividing both sides by 0.7:
n = 37.1 / 0.7
Using a calculator, n ≈ 53
So, when Un = 36.1, the term index (n) is approximately 53.