For the Arithmetic sequence, determine the value of t1, and d. An explicit formula for the general term. And T20.
7/4, 1, 1/4, -1/2
I don't know how to do this the fraction way, but by doing this the decimal way, the common difference I got is -0.75. T1 would be 7/4.
Formula would be.
Tn = a +(n-1)d
Tn = 1.75 + (n-1)-0.75
Tn = 1.75 + -0.75n -0.75
Tn = -0.75n + 1
T20....
T20 = 1.75 + (20-1)-0.75
T20 = 1.75 + (19) - 0.75
T20 = 1.75 - 14.25
T20 = -12.5
I definitely got the last one wrong, please help.
common difference = 1 - 7/4 = 4/4 - 7/4 = -3/4 (you had -.75 correct!)
check: 1/4 - 1 = 1/4 - 4/4 = -3/4
t(n) = a + (n-1)d
= 7/4 + (n-1)(-3/4)
= 7/4 - (3/4)n + 3/4
= 10/4 - (3/4)n or t(n) = 5/2 - (3/4)n
Your error was in line:
Tn = 1.75 + -0.75n -0.75
should have been
Tn = 1.75 + -0.75n +0.75
= 2.5 - .75n , the same as mine
term(20) = a + 19d
= 7/4 + 19(-3/4)
= 7/4 - 57/4
= -50/4 or - 25/2
To find the value of T20, let's use the formula for the general term of an arithmetic sequence:
Tn = a + (n-1)d
Given:
a (first term) = 7/4
d (common difference) = -3/4 (decimal value of -0.75)
Let's substitute these values into the formula to find the general term of the sequence:
Tn = (7/4) + (n-1)(-3/4)
Simplifying the equation:
Tn = 7/4 - (3/4)(n-1)
Tn = 7/4 - (3/4)(n) + 3/4
Tn = 7/4 - 3n/4 + 3/4
Tn = (7 - 3n + 3)/4
Tn = (10 - 3n)/4
Now, to find T20, substitute n = 20 into the general term equation:
T20 = (10 - 3(20))/4
T20 = (10 - 60)/4
T20 = -50/4
T20 = -25/2
So, the value of T20 is -25/2 or -12.5. You were correct in your calculation.
To determine the value of t1 and d for the arithmetic sequence, and to find an explicit formula for the general term, let's start from the given sequence: 7/4, 1, 1/4, -1/2.
To find the common difference (d), we need to look at the differences between consecutive terms. Subtract the second term from the first, the third term from the second, and so on:
(7/4 - 1) = 3/4
(1 - 1/4) = 3/4
(1/4 - (-1/2)) = 3/4
Since we obtained the same result for all differences, the common difference (d) is 3/4.
Now, to find the value of t1 (the first term), we can choose any term from the sequence. In this case, let's take the first term: 7/4.
Therefore, t1 = 7/4 and d = 3/4.
Next, we can find the explicit formula for the general term (Tn) of the arithmetic sequence. The formula is given by:
Tn = a + (n-1)d
Substituting the values we found:
Tn = (7/4) + (n-1)(3/4)
Tn = (7/4) + (3/4)(n-1)
Simplifying, we have:
Tn = (7/4) + (3/4)(n - 1)
Tn = (7/4) + (3/4)n - 3/4
Tn = (3/4)n + (4/4)
Tn = (3/4)n + 1
Now, let's find T20 (the 20th term) using the formula we just derived:
T20 = (3/4)(20) + 1
T20 = 15/4 + 1
T20 = 15/4 + 4/4
T20 = 19/4
So, T20 is equal to 19/4 or 4.75, not -12.5 as you calculated previously.