SOLUTION: In the arithmetic Sequence, the 11th term is 72(t(11)=72), and the 27th term is 96 (t(27)=96). A) Find the common Difference (generator). B)Find the initial Value, t(0). C) Writ

Algebra ->  Sequences-and-series -> SOLUTION: In the arithmetic Sequence, the 11th term is 72(t(11)=72), and the 27th term is 96 (t(27)=96). A) Find the common Difference (generator). B)Find the initial Value, t(0). C) Writ      Log On


   

Question 549647: In the arithmetic Sequence, the 11th term is 72(t(11)=72), and the 27th term is 96 (t(27)=96).
A) Find the common Difference (generator).
B)Find the initial Value, t(0).
C) Write the rule for this sequence.
Answer by KMST(5400) About Me  (Show Source):
You can put this solution on YOUR website!
The trouble here is that if your 11th term is t(11)=72, your "initial value t(0) is really your 0th term. I call the 11th term a%5B11%5D, the first term a%5B1%5D, and I do not acknowledge the existence of an initial value, or term number 0. I'll have to translate my thinking to your way of naming things.
In your way of naming terms:
The common difference is d.
There is an initial value t%280%29, but
the first term is t%281%29=t%280%29%2Bd.
The second term is .
The n-th term is t%28n%29=t%280%29%2Bnd.
The 11th term is t%2811%29=t%280%29%2B11d=72.
The 27th term is t%2827%29=t%280%29%2B27d=96.
So t%2827%29-t%2811%29=t%280%29%2B27d-%28t%280%29%2B11d%29=t%280%29%2B27d-t%280%29-11d=16d.
But t%2827%29-t%2811%29=96-72=24,
so 16d=24 --> d=24%2F16=3%2F2
The common difference is 3%2F2=1%261%2F2, and
t%2811%29=t%280%29%2B11d=t%280%29%2B11%2A%283%2F2%29=72.
So, t%280%29%2B11%2A%283%2F2%29=72 --> t%280%29=72-11%2A%283%2F2%29=72-33%2F2=144%2F2-33%2F2=111%2F2=55%261%2F2
(And the first term is t%281%29=t%280%29%2Bd=55%261%2F2%2B1%261%2F2=57).
JUST IN CASE someone whats to read this in my language
In my way of naming terms:
The common difference is d.
The first term is a%5B1%5D.
The second term is a%5B2%5D=a%5B1%5D%2Bd.
The third term is a%5B3%5D=a%5B2%5D%2Bd=a%5B1%5D%2B2d.
The n-th term is a%5Bn%5D=a%5B1%5D%2B%28n-1%29d.
The 11th term is a%5B11%5D=a%5B1%5D%2B10d=72.
The 27th term is a%5B27%5D=a%5B1%5D%2B26d=96.
So a%5B27%5D-a%5B11%5D=a%5B1%5D%2B26d-%28a%5B1%5D%2B10d%29=a%5B1%5D%2B26d-a%5B1%5D-10d=16d.
But a%5B27%5D-a%5B11%5D=96-72=24,
so 16d=24 --> d=24%2F16=3%2F2
The common difference is 3%2F2
and a%5B11%5D=a%5B1%5D%2B10d=a%5B1%5D%2B10%2A%283%2F2%29=a%5B1%5D%2B15=72 --> a%5B1%5D=72-15=57