the sum of the first nine terms of an arithmetic sequence 216. The 1st, 3rd and 7th terms form the first three terms of a geometric sequence therefore find the common difference
9/2 (2a+8d) = 216
(a+2d)/a = (1+6d)/(a+2d)
One choice is
24, 24, 24, ...
Probably not what you want. So, let's try
8, 12, 16, ...
sum(9) = (9/2)(2a + (n-1)d)
(9/2)(2a + 8d) = 216
9(a + 4d) = 216
a + 4d = 24
term1 = a
term3 = a+2d
term7 = a + 6d
they form a GP, so
(a+2d)/a = (a+6d)/(a+2d)
a^2 + 6ad = a^2 + 4d + 4d^2
2ad = 4d^2
a = 2d , assuming d ≠ 0
back into a+4d = 24
2d + 4d = 24
d = 4
a = 8
check:
terms are
8 , 12, 16, 20 , 24, 28, 32...
term7/term3 = term3/term1
32/16 = 16/8
true!
sum(9) = (9/2)(16 + 8(4)) = 216 , ok
To find the common difference of an arithmetic sequence, we need to use the given information that the sum of the first nine terms is 216.
The formula to find the sum of an arithmetic sequence is:
Sum = (n/2)(2a + (n-1)d)
Where:
- Sum is the sum of the first n terms,
- n is the number of terms,
- a is the first term, and
- d is the common difference.
In this case, we are given that the sum of the first nine terms is 216. So we can write the equation as:
216 = (9/2)(2a + (9-1)d)
To simplify further, we can multiply both sides of the equation by 2:
432 = 9(2a + 8d)
Expanding and rearranging the equation:
432 = 18a + 72d
Now, we are also given that the 1st, 3rd, and 7th terms form the first three terms of a geometric sequence. In a geometric sequence, the ratio between consecutive terms is constant.
Let's find the common ratio of this geometric sequence using the given information:
1st term = a
3rd term = a + 2d
7th term = a + 6d
Since the ratio between consecutive terms is constant, we can write:
(a + 2d) / a = (a + 6d) / (a + 2d)
Cross multiply to solve:
(a + 2d)(a + 2d) = (a + 6d)(a)
Simplifying this equation:
a^2 + 4ad + 4d^2 = a^2 + 6ad
Subtracting a^2 from both sides:
4ad + 4d^2 = 6ad
Simplifying further:
2ad + 4d^2 = 0
Factoring out d:
d(2a + 4d) = 0
Now, we have two possibilities:
1. d = 0, which means the arithmetic sequence has a common difference of 0, and therefore all terms are the same. However, this would not form a valid geometric sequence with distinct terms.
2. 2a + 4d = 0, which implies 2a = -4d, and simplifying it further, a = -2d.
Substituting a = -2d into the equation 432 = 18a + 72d:
432 = 18(-2d) + 72d
Simplifying this equation:
432 = -36d + 72d
Combining like terms:
432 = 36d
Dividing both sides by 36:
12 = d
Therefore, the common difference of the arithmetic sequence is 12.