SOLUTION: If tn = 10/3 − n/3 , find t1, t2, t3 and tn+1 . Express tn+1−tn in its simplest form.

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Question 1150467: If tn = 10/3 − n/3 , find t1, t2, t3 and tn+1 . Express tn+1−tn in its simplest form.
Found 2 solutions by greenestamps, Theo:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Given: t%28n%29+=+10%2F3-n%2F3+=+%2810-n%29%2F3

The alternate form will make it easier to find specified terms of the sequence.

t(1): t%281%29+=+%2810-1%29%2F3+=+9%2F3+=+3

t(2), t(3): You can do those....

t(n+1): t%28n%2B1%29+=+%2810-%28n%2B1%29%29%2F3+=+%2810-n-1%29%2F3+=+%289-n%29%2F3

t(n+1)-t(n):


Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
T[n] = 10/3 - n/3.

T[1] = 10/3 - 1/3 = 9/3

T[2] = 10/3 - 2/3 = 8/3

T[3] = 10/3 - 3/3 = 7/3

T[n+1] = 10/3 - (n+1)/3

T[n+1] - T[n] = 10/3 - (n+1)/3 - (10/3 - n/3)
simplify this to get:
T[n+1] - T[n] = 10/3 - (n+1)/3 - 10/3 + n/3
combine like terms to get:
T[n+1] - T[n] = -(n+1)/3 + n/3 = (-n - 1 + n) / 3 = -1/3

for example:

T3 = 10/3 - 3/3 = 7/3
T2 = 10/3 - 2/3 = 8/3

if you let T[n] = T[2] and, if you let T[n+1] = T[3], then you get:
T[n+1] - T[n] = T[3] - T[2] = (10/3 - 3/3) - (10/3 - 2/3) = 7/3 - 8/3 = -1/3.

this is correct according to the formula, as far as i can tell.

further, if you look at T[1] and T[2] and T[3], you will see that each succeeding value is 1/3 less than the preceding value by 1/3.