1.1 the third term of an arithmetic sequence is 8 and the 15 term is 44 . Calculate: 1.1 the common difference and first term. 1.2 the sum of the first 50 terms. 1.2 1;4;7;10.... Is an arithmetic sequence .find: 1.2.1 the 30 term.
https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
http://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html
might work better
1.1 To find the common difference and the first term of an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.
Given that the third term (a3) is 8 and the fifteenth term (a15) is 44, we can set up two equations:
a3 = a1 + (3 - 1)d -> 8 = a1 + 2d ...........(1)
a15 = a1 + (15 - 1)d -> 44 = a1 + 14d ...........(2)
We have a system of equations to solve. Let's solve them simultaneously:
First, subtract equation (1) from equation (2), to eliminate a1:
44 - 8 = 14d - 2d
36 = 12d
d = 36/12 = 3
Now, substitute the value of d = 3 into equation (1) and solve for a1:
8 = a1 + 2(3)
8 = a1 + 6
a1 = 8 - 6
a1 = 2
Therefore, the common difference (d) is 3 and the first term (a1) is 2.
1.2 To find the sum of the first 50 terms of an arithmetic sequence, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a1 + (n - 1)d)
where Sn is the sum of the first n terms, a1 is the first term, n is the number of terms, and d is the common difference.
Plugging in the values we know:
a1 = 2 (from previous calculation)
d = 3 (from previous calculation)
n = 50
Substitute these values into the formula and calculate:
S50 = (50/2)(2(2) + (50 - 1)(3))
= 25(4 + 49(3))
= 25(4 + 147)
= 25(151)
= 3775
Therefore, the sum of the first 50 terms is 3775.
1.2.1 The given sequence 1; 4; 7; 10... is an arithmetic sequence with a common difference of 3. To find the 30th term, we can use the same formula as before:
an = a1 + (n - 1)d
Plugging in the values we know:
a1 = 1 (the first term)
d = 3 (the common difference)
n = 30
Substitute these values into the formula and calculate:
a30 = 1 + (30 - 1)3
= 1 + 29(3)
= 1 + 87
= 88
Therefore, the 30th term of the sequence is 88.