The following problem refers to an arithmetic sequence. If the seventh term is 9 and the fourteenth term is 16, find the term a1, the common difference
d, and then find a27 and S27
since there are 7 terms from a7 to a14, and the difference is 7, d=1
so, a1=3
Now you can easily find a27, and of course,
S27 = 27/2 (a1+a27))
To solve this problem, we will use the formulas for arithmetic sequences.
In an arithmetic sequence, the nth term (an) can be found using the formula:
an = a1 + (n-1) * d
where a1 is the first term, n is the position term, and d is the common difference.
Given that the seventh term (a7) is 9 and the fourteenth term (a14) is 16, we can set up two equations using the formula above:
a7 = a1 + 6d (Equation 1)
a14 = a1 + 13d (Equation 2)
To find a1 and d, we can solve these two equations simultaneously.
Subtracting Equation 1 from Equation 2, we have:
a14 - a7 = (a1 + 13d) - (a1 + 6d)
16 - 9 = 13d - 6d
7 = 7d
Dividing both sides by 7, we find:
d = 1
Substituting the value of d into Equation 1, we have:
9 = a1 + 6(1)
a1 = 9 - 6
a1 = 3
So, we have found that the first term, a1, is 3 and the common difference, d, is 1.
To find the 27th term (a27), we can use the formula:
a27 = a1 + (27-1) * d
a27 = 3 + 26 * 1
a27 = 3 + 26
a27 = 29
The 27th term (a27) is 29.
To find the sum of the first 27 terms (S27), we can use the formula for the sum of an arithmetic sequence:
S27 = (n/2) * (a1 + an)
Substituting the values, we have:
S27 = (27/2) * (3 + 29)
S27 = 13.5 * 32
S27 = 432
The sum of the first 27 terms (S27) is 432.