The fifth term of an arithmetic progression is 19 and the fourteenth term is 15. Find the, first term, common difference, and sum of the first eighty terms.
remember that t(n) = a+(n-1)d
t(8) = a+7d
t(4) = a+3d
it said: the 8th term is twice the 4th term ---> a+7d = 2(a+3d), simplify this
then it said: the 20th term is 40 ---> a+19d = 40
Solve these two equations.
To find the first term (a) and the common difference (d) of an arithmetic progression (AP), we can use the formula:
aₙ = a + (n - 1)d
Where aₙ is the nth term of the AP, a is the first term, n is the position of the term in the AP, and d is the common difference.
Given that the fifth term (a₅) is 19 and the fourteenth term (a₁₄) is 15, we can set up the following equations:
a₅ = a + (5 - 1)d [Equation 1]
a₁₄ = a + (14 - 1)d [Equation 2]
Substituting the known values into the equations:
19 = a + 4d [Equation 1]
15 = a + 13d [Equation 2]
To solve these equations, we can use the method of elimination or substitution. Let's use the method of substitution.
From Equation 1, we can express a in terms of d:
a = 19 - 4d [Equation 3]
Substituting Equation 3 into Equation 2:
15 = (19 - 4d) + 13d
Simplifying:
15 = 19 - 4d + 13d
15 = 19 + 9d
9d = 15 - 19
9d = -4
d = -4/9
Now we can substitute the value of d (-4/9) into Equation 3 to find the value of a:
a = 19 - 4d
a = 19 - 4(-4/9)
a = 19 + (16/9)
a = 19 + 16/9
a = (171 + 16)/9
a = 187/9
Therefore, the first term (a) is 187/9 and the common difference (d) is -4/9.
To find the sum of the first eighty terms of the AP, we can use the formula for the sum of an AP:
Sₙ = (n/2)(2a + (n - 1)d)
Where Sₙ is the sum of the first n terms, a is the first term, n is the number of terms, and d is the common difference.
Substituting the known values:
S₈₀ = (80/2)(2(187/9) + (80 - 1)(-4/9))
S₈₀ = 40(374/9 - 79(4/9))
S₈₀ = 40(374/9 - 316/9)
S₈₀ = 40(58/9)
S₈₀ = 40 * 58 / 9
S₈₀ = 2320/9
Therefore, the sum of the first eighty terms of the arithmetic progression is 2320/9.