SOLUTION: Please help me solve this: A certain arithmetic sequence has {{{a[51]=23}}} and {{{a[101]=-127}}}. Find {{{a[1]}}}, {{{d}}}, {{{a[n]}}}, and {{{a[78]}}}

Algebra ->  Sequences-and-series -> SOLUTION: Please help me solve this: A certain arithmetic sequence has {{{a[51]=23}}} and {{{a[101]=-127}}}. Find {{{a[1]}}}, {{{d}}}, {{{a[n]}}}, and {{{a[78]}}}      Log On


   



Question 323764: Please help me solve this: A certain arithmetic sequence has a%5B51%5D=23
and a%5B101%5D=-127. Find a%5B1%5D, d, a%5Bn%5D, and a%5B78%5D

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me solve this: A certain arithmetic sequence has a%5B51%5D=23
and a%5B101%5D=-127. Find a%5B1%5D, d, a%5Bn%5D, and a%5B78%5D

Use  a%5Bn%5D%22%22=%22%22a%5B1%5D%2B%28n-1%29d%29

Substitute n=51, a%5B51%5D=23

a%5B51%5D%22%22=%22%22a%5B1%5D%2B%2851-1%29d%29
23%22%22=%22%22a%5B1%5D%2B50d%29
a%5B1%5D%2B50d%29%22%22=%22%2223

Substitute n=101, a%5B101%5D=-127

a%5B101%5D%22%22=%22%22a%5B1%5D%2B%28101-1%29d%29
-127%22%22=%22%22a%5B1%5D%2B100d%29
a%5B1%5D%2B100d%29%22%22=%22%22-127


So we have this system of equations:

system%28a%5B1%5D%2B50d=23%2C+a%5B1%5D%2B100d=-127%29

Solve that system of equation by substitution or addition.
If you don't know how post again asking how.  The
solution is a%5B1%5D%22%22=%22%22173, d""=""}}}-3

Therefore substituting those in

a%5Bn%5D%22%22=%22%22a%5B1%5D%2B%28n-1%29d%29
a%5Bn%5D%22%22=%22%22173%2B%28n-1%29%28-3%29
a%5Bn%5D%22%22=%22%22173%2B%28-3%29%28n-1%29%29
a%5Bn%5D%22%22=%22%22173-3%28n-1%29%29
a%5Bn%5D%22%22=%22%22173-3n%2B3%29%29
a%5Bn%5D%22%22=%22%22176-3n%29

Substitute n=78
a%5Bn%5D%22%22=%22%22176-3n%29
a%5B78%5D%22%22=%22%22176-3%2878%29%29
a%5B78%5D%22%22=%22%22176-234%29
a%5B78%5D%22%22=%22%22-58%29

Edwin