The first three terms of an A.P are x, (3x+1), (7x-4) .find (1) the value of a
(2) 10th term
Well, if the first term is x, the second term is (3x+1), and the third term is (7x-4), then we can use this information to find the common difference, d, of the arithmetic progression.
To find the common difference, we subtract the second term from the first term:
(3x+1) - x = 2x + 1
And we subtract the third term from the second term:
(7x-4) - (3x+1) = 4x - 5
Now, since it's an arithmetic progression, the common difference should be the same. So, we can equate these two expressions:
2x + 1 = 4x - 5
Now let's solve this equation to find the value of x, which will help us find the common difference, a!
2x - 4x = -5 - 1
-2x = -6
x = 3
So, x = 3, which means the first term is 3.
Now, to find the common difference, we substitute x = 3 into either 2x + 1 or 4x - 5:
2(3) + 1 = 7
4(3) - 5 = 7
Hurray! The common difference is 7.
Now, we can use this information to find the 10th term of the arithmetic progression. The formula for the nth term of an arithmetic progression is:
a(n) = a + (n-1)d
where a is the first term and d is the common difference.
Let's substitute the values we know:
a(10) = 3 + (10-1)7
= 3 + 9 * 7
= 3 + 63
= 66
Therefore, the 10th term is 66.
I hope my math didn't get too clowny!
In A.P
a = initial term
d = common difference
n-th term:
an = a + ( n - 1 ) d
a1 = a
a2 = a + d
a3 = a + 2 d
In this case
a1 = x
a2 = 3 x + 1
a3 = 7 x - 4
so:
a1 = x
a2 = a1 + d
3 x + 1 = x + d
a3 = a1 + 2 d
7 x - 4 = x + 2 d
Now you must solve system:
3 x + 1 = x + d
7 x - 4 = x + 2 d
First equation.
3 x + 1 = x + d
Subtract x to both sides.
2 x + 1 = d
d = 2 x + 1
Second equation.
7 x - 4 = x + 2 d
Subtract x to both sides.
6 x - 4 = 2 d
2 d = 6 x - 4
Divide both sides by 2.
d = 3 x - 2
Now:
d = d
2 x + 1 = 3 x - 2
Subtract 2x to both sides.
1 = x - 2
Add 2 to both sides.
3 = x
x = 3
d = 2 x + 1
d = 2 • 3 + 1 = 6 + 1
d = 7
Or
d = 3 x - 2
d = 3 • 3 - 2 = 9 - 2
d = 7
1)
a = a1 = x
a = 3
2)
a10 = a1 + 9 d
a10 = 3 + 9 ∙ 7 = 3 + 63 = 66
Your A.P.
3 , 10 , 17 , 24 , 31 , 38 , 45 , 52 , 59 , 66 ...
To find the value of the first term (a) in the arithmetic progression (AP), we can equate it with the given value of x.
1) Value of a (first term):
a = x
To find the 10th term of the AP, we need to find the common difference (d) first. The common difference can be found by subtracting the second term from the first term (as they are consecutive terms in the AP).
Common difference (d):
d = (3x + 1) - x
d = 2x + 1
Now we can use the formula to find the nth term of an AP:
nth term (Tn) = a + (n-1)d
2) 10th term:
T10 = a + (10-1)d
T10 = x + (9)(2x + 1)
Simplifying the expression:
T10 = x + 18x + 9
T10 = 19x + 9
Therefore, the value of the 10th term is 19x + 9.
To find the value of the first term (a) of an arithmetic progression (AP), we need to know the common difference (d) between the terms. In this case, we can find the common difference by taking the difference between the second and first terms, or the difference between the third and second terms.
Let's find the common difference:
Common difference (d) = (3x + 1) - x
= 2x + 1
Now that we know the common difference, we can find the value of the first term (a) by substituting it into the given expression for the first term:
x = a
Therefore, the first term (a) is equal to x.
To find the 10th term of the AP, we can use the formula:
nth term (Tn) = a + (n-1)d
Given that the first term (a) is x and the common difference (d) is 2x + 1, we can substitute these values into the formula:
10th term (T10) = x + (10-1)(2x + 1)
= x + 9(2x + 1)
= x + 18x + 9
= 19x + 9
Therefore, the 10th term of the AP is 19x + 9.