The 10th term of an ap is-27 and the 5th term is -12 what is the 18th find also the sum of its 25th term
a+9d = -27
a+4d = -12
subtract:
5d = -15
d = -3
sub into a + 4d = -12 to find a, then use your definitions to find the rest of question.
btw, "..the sum of its 25th term" makes no sense
In AP n-th term is:
an = a + ( n - 1 ) d
where
a = first term
d = common difference
In this case:
a10 = a + 9 d
a10 = - 27
a + 9 d = - 27
a5 = a + 4 d
a5 = - 12
a + 4 d = - 12
Now you must solve system of two equations:
a + 9 d = - 27
a + 4 d = - 12
The solution is:
a = 0 , d = - 3
a18 = a + 17 d
a18 = 0 + 17 ∙ ( - 3 )
a18 = - 51
Sum of first n members of arithmetic progression is:
Sn = n [ 2 a + ( n - 1 ) d ] / 2
S25 = 25 ∙ [ 2 ∙ 0 + 24 ∙ ( - 3 ) ] / 2 = 25 ∙ ( 0 - 72 ) / 2 =
25 ∙ ( - 72 ) / 2 = - 1800 / 2
S25 = - 900
To find the 18th term of the arithmetic progression (AP), we can use the formula:
aₙ = a₁ + (n - 1)d
Where:
aₙ is the nth term of the AP,
a₁ is the first term of the AP,
n is the term number,
d is the common difference between consecutive terms.
Given information:
a₅ = -12
a₁₀ = -27
Step 1: Finding the common difference (d)
We can find the common difference by subtracting the 5th term from the 10th term and dividing by the difference in term numbers.
d = (a₁₀ - a₅) / (10 - 5)
d = (-27 - (-12)) / 5
d = -15 / 5
d = -3
Step 2: Finding the first term (a₁)
Using the formula for the nth term of an AP and the known values, we can find the first term:
a₅ = a₁ + (5 - 1)(-3)
-12 = a₁ - 12
a₁ = 0
Step 3: Finding the 18th term (a₁₈)
Using the formula for the nth term of an AP and the known values, we can find the 18th term:
a₁₈ = a₁ + (18 - 1)(-3)
a₁₈ = 0 + 17(-3)
a₁₈ = 0 - 51
a₁₈ = -51
Therefore, the 18th term of the AP is -51.
Step 4: Finding the sum of the 25th term
To find the sum of the 25th term, we can use the formula for the sum of an AP:
Sₙ = (n/2)(a₁ + aₙ)
Where:
Sₙ is the sum of the first n terms of the AP.
Using this formula, we can calculate the sum of the 25th term:
S₂₅ = (25/2)(0 + a₂₅)
However, we don't have the 25th term (a₂₅) given. Please provide the value of the 25th term, so we can calculate the sum.
To find the 10th term of an arithmetic progression (AP), we first need to find the common difference (d) of the AP.
Given that the 5th term is -12 and the 10th term is -27, we can use the formula for the nth term of an AP:
a(n) = a(1) + (n-1)d
where a(n) is the nth term, a(1) is the first term, n is the term position, and d is the common difference.
Using the formula, we can substitute the values we know:
-12 = a(1) + (5-1)d
-27 = a(1) + (10-1)d
From equation (1):
-12 = a(1) + 4d
From equation (2):
-27 = a(1) + 9d
Subtracting equation (1) from equation (2) to eliminate a(1):
-27 - (-12) = a(1) + 9d - (a(1) + 4d)
-27 + 12 = 9d - 4d
-15 = 5d
Dividing both sides by 5 to solve for d:
-15/5 = d
-3 = d
Now that we have the common difference (d = -3), we can find the 18th term using the formula:
a(n) = a(1) + (n-1)d
Substituting the known values:
a(18) = a(1) + (18-1)(-3)
a(18) = a(1) + 17(-3)
a(18) = a(1) - 51
Therefore, the 18th term of the arithmetic progression is a(18) = a(1) - 51.
To find the sum of the 25th term, we can use the formula for the sum of n terms of an AP:
S(n) = (n/2)(a(1) + a(n))
Substituting the known values:
S(25) = (25/2)(a(1) + a(25))
We already know a(1) = -12, and to find a(25), we can use the formula mentioned earlier:
a(25) = a(1) + (25-1)d
a(25) = a(1) + 24d
a(25) = -12 + 24(-3)
a(25) = -12 - 72
a(25) = -84
Now we can substitute the values into the sum formula:
S(25) = (25/2)(-12 + -84)
S(25) = (25/2)(-96)
S(25) = (25 * -96) / 2
S(25) = -2400/2
S(25) = -1200
Therefore, the sum of the 25th term of the arithmetic progression is -1200.