SOLUTION: The ninth and seventeenth terms of an arithmetic sequence are negative 8 and 32​, respectively. Find the first term and a recursive rule for the nth term.

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Question 1136793: The ninth and seventeenth terms of an arithmetic sequence are negative 8 and 32​, respectively. Find the first term and a recursive rule for the nth term.
Found 3 solutions by ikleyn, rothauserc, greenestamps:
Answer by ikleyn(52847)   (Show Source): You can put this solution on YOUR website!
.
Let "a" be the first term of the progression and  "d" be the common difference.


Then from the condition you have these 2 equations


 = a + 8d = -8

 = a + 16d = 32.


At this point I will stop my writing, since from your post it is not clear to me if 32 is positive or negative;
and I don't want to spend my time for nothing.

Learn how to write your posts in unambiguous way.

In this case you could easily write "-8" instead of saying "negative 8".

Do you know where the minus sign " - " is located in your keyboard ?


Keep in mind that you lost a rare and happy opportunity to learn from me, since you presented your post in inaccurate way.


/\/\/\/\/\/\/\/

Regarding the post by @rothauserc,  keep in mind that  EVERYTHERE  where he writes  " a(0) ",  it should be read as " a(1) ".



Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
the formula for the nth term of an arithmetic sequence is
:
a(n) = a(0) +d(n-1), where a(0) is the first term and d is the common difference
:
we are given two equations in two unknowns
:
a(9) = -8 = a(0) +d(9-1)
:
1) a(0) +8d = -8
:
a(17) = 32 = a(0) +d(17-1)
:
2) a(0) +16d = 32
:
solve equation 1 for a
:
a(0) = -8 -8d
:
substitute for a(0) in equation 2
:
(-8 -8d) +16d = 32
:
8d = 40
:
d = 5
:
a(0) = -8 -8(5)
:
a(0) = -48
:
****************************************************
the first term is -48
:
the recursive rule is
:
a(0) = -48, a(n) = a(n-1) +5
:
check the answer
:
For equation 1
:
a(9) = -48 +8(5) = -8
:
-8 = -8
:
For equation 2
:
a(17) = -48 +16(5) = 32
:
32 = 32
:
answer checks
*****************************************************
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


The problem is relatively easy to solve informally, without the algebraic formulas regarding arithmetic sequences.

While you probably want to learn and understand those formulas, solving problems like this using logical reasoning and simple mental calculations is good mental exercise.

The difference between the 9th and 17th terms of an arithmetic sequence is 8 times the common difference. The difference between the 9th and 17th terms is 32-8=24; so the common difference is 24/8 = 3.

The difference between the 1st and 9th terms is also 8 times the common difference; so the first term is 8-24 = -16.

With a common difference of 3, a recursive rule for the n-th term is

t(n) = t(n-1)+3
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