How do I renumber a sequence? (whether arithmetic or geometric) Without being given the first term?

Question

Such as in an arithmetic problem if youre given, t5= 94, t12=45 and it asks you to find t1, how would you do that?

Answer 1

If it is a arithmetic sequence,

Given t5 = 94 then a+4d = 94

t12 = 45 then a+11d = 45

Subtract t12 from t5.

a+4d   = 94

a+11d = 45

(-) (-)     (-)

___________

-7d = 49

Divide by negitive 7 to each side.

-7d/-7 = 49/-7

d = - 7

Substitute the d value in t5.

a+4*-7=94

a-28 = 94

Add 28 from each side.

a-28+28 = 94+28

a =122.

To check is it arithemetic sequence?.

Given t5 = 94

a+4d = 94

Substitute the values of a,d.

122+4*-7 = 94

122-28 = 94

94 = 94

So it is arithemetic sequence.