find the explicit formula
a7=-10 and a13=-48
The problem is to find a formula that we can use to
fill in the empty blanks of this sequence:
___,___,___,___,___,___,-10,___,___,___,___,___,-48
Let's assume it's an arithmetic sequence. Then we use
an = a1 + (n-1)d
Substitute n=7
an = a1 + (7-1)d
a7 = a1 + 6d
when n=7, an = -10, so a7 = -10
Substituting a7 = -10
-10 = a1 + 6d
Also,
an = a1 + (n-1)d
Substitute n=13
a13 = a1 + (13-1)d
a13 = a1 + 12d
when n=13, an = -48, so a13 = -48
Substituting a13 = -48
-48 = a1 + 12d
So we have the system of two equations and
two unknowns
-10 = a1 + 6d
-48 = a1 + 12d
So we solve that system and get:
a1 = 28, d = -19/3
So to get the formula, we take
an = a1 + (n-1)d
And substitute those values of a1 and d
an = 
an = 
an = 
an = 
That's the formula. We can now fill in the blanks:
28,65/3,46/3,9,8/3,-11/3,-10,-49/3,-68/3,-29,-106/3,-48.
-------------------------------------
That's the solution your teacher wants, most likely.
However, it is also possible that it could be a
geometric sequence as well:
Let's assume it's a geometric sequence. Then we use
an = a1rn-1
Substitute n=7
a7 = a1r7-1
a7 = a1r6
when n=7, an = -10, so a7 = -10
Substituting a7 = -10
-10 = a1r6
Also,
an = a1rn-1
Substitute n=13
a13 = a1r13-1
a13 = a1r12
when n=13, an = -48, so a13 = -48
Substituting a13 = -48
-48 = a1r12
So we have the system of two equations and
two unknowns
-10 = a1r6
-48 = a1r12
So we solve that system and get:
a1 = 
, d = 
So to get the formula, we take
an = a1rn-1
And substitute those values of a1 and d
an = 
That's the formula. We can now fill in the blanks,
but I'll just give their decimal values:
-2.083,-2.706,-3.514,-4.564,-5.928,-7.699,
-10,-12.99,-16.87,-21.91,-28.46,-36.96,-48.
Actually, arithmetic and geometric sequences are only 2 kinds of
many sequences. So technically there are infinitely many answers to
this problem, but only one arithmetic sequence solution, and only one
geometric sequence solution.
Edwin
