The first term of a. p is 7 and the last term is 20. the sum of the terms is252.calculate common difference and the number of terms
Of course the bot can't do such a "complicated" problem, and got it wrong.
given: a= 7
--- term(n) = 20
a+ (n-1)d = 20
(n-1)d = 13
sum(n) = (n/2)(2a + (n-1)d )
252 = (n/2)(14 + 13)
504 = 27n
n = 72
in (n-1)d = 13
71d = 13
d = 13/71
I apologize for the mistake in the previous response. Thank you for correcting it and providing the correct solution.
To find the common difference and the number of terms in an arithmetic progression (AP), we can use the following formulas:
1. The formula for the nth term of an AP is:
a + (n - 1)d = Tn,
where 'a' is the first term, 'n' is the term number, 'd' is the common difference, and 'Tn' is the nth term.
2. The sum of the first 'n' terms of an AP is given by the formula:
Sn = (n/2)(2a + (n-1)d),
where 'Sn' is the sum of the 'n' terms.
Let's use these formulas to solve the problem:
Given:
First term (a) = 7
Last term = 20
Sum of terms (Sn) = 252
Step 1: Finding the number of terms (n)
We need to find the value of 'n' using the formula for the sum of 'n' terms.
Sn = (n/2)(2a + (n-1)d), substitute the given values:
252 = (n/2)(2*7 + (n-1)d)
252 = (n/2)(14 + (n-1)d)
Simplify the equation:
252 = 7n + (n^2 - n)d
252 = 7n + nd^2 - nd
Step 2: Finding the common difference (d)
Since we have two variables 'n' and 'd' in the equation, we need another equation to solve for 'd'.
To find the common difference, we can use the formula for the nth term:
a + (n - 1)d = Tn
Substituting the given values:
7 + (n - 1)d = 20
Step 3: Solving the equations simultaneously
Now, we have two equations:
252 = 7n + nd^2 - nd .....(Equation 1)
7 + (n - 1)d = 20 .....(Equation 2)
From Equation 2, we can rewrite it as:
(n - 1)d = 20 - 7
(n - 1)d = 13
We can now substitute (n - 1)d = 13 into Equation 1:
252 = 7n + 13n - 13
252 = 20n - 13
Simplify:
20n = 252 + 13
20n = 265
Divide both sides by 20:
n = 265 / 20
n = 13.25
Since 'n' represents the number of terms, it should be a whole number. Therefore, the number of terms is approximately 13.
To find the common difference, substitute the value of 'n' into Equation 2:
7 + (13 - 1)d = 20
7 + 12d = 20
12d = 20 - 7
12d = 13
d = 13 / 12
d = 1.08 (approximately)
Therefore, the common difference is approximately 1.08 and the number of terms is approximately 13.
Let the common difference be d and the number of terms be n.
We know that the nth term of an A.P. is given by:
an = a1 + (n-1)d, where a1 is the first term.
Given, a1 = 7, an = 20, and the sum of the terms is 252.
So, we have:
an = a1 + (n-1)d
=> 20 = 7 + (n-1)d
=> 13 = (n-1)d (1)
Also, we have the formula for the sum of n terms of an A.P.:
Sn = n/2 [2a1 + (n-1)d]
Substituting the given values, we get:
252 = n/2 [2(7) + (n-1)d]
=> 252 = n/2 [14 + (n-1)d]
=> 18 = n/2 (1 + (n-1)d/7) (2)
From equations (1) and (2), we can substitute (n-1)d as 13 and simplify:
18 = n/2 (1 + 13/7)
=> 18 = n/2 (20/7)
=> n = 36/5 = 7.2
However, n must be a whole number. This means that we need to round off 7.2 to the nearest whole number. We can do this by considering whether the decimal part is greater than or equal to 0.5. In this case, 0.2 is less than 0.5. So we round down to n = 7.
Now that we have the value of n, we can substitute it in equation (1) to find d:
13 = (n-1)d
=> 13 = 6d
=> d = 13/6
Therefore, the common difference is 13/6 and the number of terms is 7.