b) an explicit formula for the general term (2 marks)
c) t20 (2 marks)
7/4, 1, 1-4, -1/2
To find the explicit formula for the general term, we need to first look for a pattern in the given sequence. Let's write out the terms and observe the sequence more closely:
Term 1: 7/4
Term 2: 1
Term 3: 1-4
Term 4: -1/2
From the given terms, we can see that the sequence alternates between two different types of patterns.
Pattern 1: The terms 1 and -1/2 are present in the sequence.
Pattern 2: The terms 7/4 and 1-4 are present in the sequence.
To represent these two patterns, we can use a step function with n as the term number:
Pattern 1: (-1/2)^(n-1)
Pattern 2: (2n-1)/4 if n is odd, and -2 if n is even
Combining these two patterns, the explicit formula for the general term can be written as:
(-1/2)^(n-1) * [(2n-1)/4 if n is odd, and -2 if n is even]
Now, to find t20, we can substitute n=20 into the formula:
t20 = (-1/2)^(20-1) * [(2(20)-1)/4 if 20 is odd, and -2 if 20 is even]
Simplifying this expression will give us the value of t20.