The fourth and ninth term of an a.p are -3 and 12 repectively find:the common different and the sum of the first seven term.
a+3d = -3
a+8d = 12
subtract:
5d = 15
d = 15/5 = 3
a + 9 = -3
a = -12
common difference is 3
sum(7) = (7/2)(-24 + 6(3)) = -21
Why did the arithmetic progression go to therapy?
It needed help finding its common difference.
But don't worry, I'm here to lend a hand! Let's solve this together.
Given that the fourth term (a₄) is -3 and the ninth term (a₉) is 12, we can find the common difference (d) between the terms.
We know that the formula for the nth term is aₙ = a₁ + (n-1)d. Let's substitute in the values using a₄ and a₉:
a₄ = a₁ + (4-1)d (1)
a₉ = a₁ + (9-1)d (2)
We have two equations with two unknowns (a₁ and d). By solving these simultaneous equations, we can find the common difference and then calculate the sum of the first seven terms.
Subtract equation (1) from equation (2) to eliminate a₁:
a₉ - a₄ = a₁ + (9-1)d - [a₁ + (4-1)d]
12 - (-3) = 8d
15 = 8d
d = 15/8
So, the common difference (d) is 15/8.
Now, let's find the sum of the first seven terms (S₇) using the formula:
S₇ = (n/2)(2a₁ + (n-1)d)
Plugging in the values, we have:
S₇ = (7/2)(2a₁ + (7-1)(15/8))
= (7/2)(2a₁ + 6(15/8))
= (7/2)(2a₁ + 45/4)
= (7/1)(a₁ + 45/8)
= 49a₁/8 + 315/8
Unfortunately, without the first term (a₁), we cannot find the precise sum of the first seven terms. If you could provide the value of the first term, I'd be more than happy to calculate the result for you!
To find the common difference of an arithmetic progression (a.p.), we can use the formula:
Common difference (d) = nth term - (n-1)th term
Given that the fourth term is -3 and the ninth term is 12, we can find the common difference as follows:
d = 12 - (-3) [Substitute values into the formula]
d = 12 + 3
d = 15
So, the common difference of the arithmetic progression is 15.
To find the sum of the first seven terms of the arithmetic progression, we can use the formula for the sum of an arithmetic series:
Sum of first n terms (Sn) = n/2 * (2a + (n-1)d)
In this case, we have:
n = 7 [Number of terms]
a = -3 [First term]
d = 15 [Common difference]
Substituting these values into the formula, we can find the sum of the first seven terms:
Sn = 7/2 * (2*(-3) + (7-1)*15)
Sn = 7/2 * (-6 + 6*15)
Sn = 7/2 * (-6 + 90)
Sn = 7/2 * 84
Sn = 7 * 42
Sn = 294
Therefore, the sum of the first seven terms of the arithmetic progression is 294.
To find the common difference of an arithmetic progression (AP), you need to subtract any two consecutive terms from each other. In this case, we subtract the fourth term (-3) from the ninth term (12).
Common Difference = 12 - (-3)
Common Difference = 15
So, the common difference of the arithmetic progression is 15.
To find the sum of the first seven terms of an AP, we can use the formula for the sum of an arithmetic series:
Sum of First n Terms (Sn) = (n/2) * [2a + (n - 1)d]
Where n represents the number of terms, a represents the first term, and d represents the common difference.
In this case, we want to find the sum of the first seven terms (n = 7). The first term (a) is given as -3, and we already found the common difference (d) to be 15.
Substituting these values into the formula:
Sn = (7/2) * [2*(-3) + (7 - 1) * 15]
= (7/2) * [-6 + 6 * 15]
= (7/2) * [-6 + 90]
= (7/2) * 84
= 7 * 42
= 294
Therefore, the sum of the first seven terms of the arithmetic progression is 294.