The fourth and sixth term of an arithmetic progession are -4 and -10 respectively. Find
A.the common difference
B.the first term
C.sum of the first twelve terms
d is easy, since A6-A4 = 2d, so
2d = -6
Now you can find a, since a+2d = -4
S12 = 12/2 (2a+11d)
To find the common difference (d) of an arithmetic progression, you can subtract any two consecutive terms. Let's subtract the sixth term (-10) from the fourth term (-4).
-10 - (-4) = -10 + 4 = -6
Therefore, the common difference (d) is -6.
To find the first term (a₁) of the arithmetic progression, you can use the formula:
aₙ = a₁ + (n - 1)d
where aₙ is the nth term, n is the position of the term, and d is the common difference.
We are given that the sixth term (a₆) is -10. Let's use this information to find the first term (a₁).
a₆ = a₁ + (6 - 1)(-6)
-10 = a₁ + 5(-6)
-10 = a₁ - 30
a₁ = -10 + 30
a₁ = 20
Therefore, the first term (a₁) is 20.
To find the sum of the first twelve terms (S₁₂) of an arithmetic progression, you can use the formula:
Sₙ = (n/2)(2a₁ + (n - 1)d)
where Sₙ is the sum of the first n terms, n is the number of terms, a₁ is the first term, and d is the common difference.
Using this formula, let's find the sum of the first twelve terms (S₁₂).
S₁₂ = (12/2)(2(20) + (12 - 1)(-6))
S₁₂ = 6(40 + 11(-6))
S₁₂ = 6(40 - 66)
S₁₂ = 6(-26)
S₁₂ = -156
Therefore, the sum of the first twelve terms (S₁₂) is -156.