How many terms of the AP 17,15,13 are need to give the sum 72.explain the double answer
To find the number of terms of the AP 17,15,13 needed to give a sum of 72, we can use the formula for the sum of the first n terms of an AP:
S_n = n/2(2a + (n-1)d)
where S_n is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
Substituting the values given in the question, we get:
72 = n/2(2(17) + (n-1)(-2))
Simplifying this expression, we get:
72 = n(17 - n + 1)
72 = n(18 - n)
Expanding the right side of the equation, we get:
72 = 18n - n^2
Rearranging this equation and setting it equal to zero, we get:
n^2 - 18n + 72 = 0
Factoring this quadratic equation, we get:
(n-6)(n-12) = 0
Therefore, the possible values for n are 6 and 12.
There are two possible answers because the sequence 17,15,13 can either be increasing (-2 common difference) or decreasing (2 common difference). If the sequence is increasing, we need 12 terms to reach the sum of 72, but if the sequence is decreasing, we only need 6 terms to reach the same sum.
Solve x:1+4+7+10+.....+x=590
Notice that the given sum is an arithmetic series with first term 1, common difference 3, and last term x. We can find x by using the formula for the sum of an arithmetic series:
S = n/2(2a + (n-1)d)
where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
Substituting the known values, we have:
590 = n/2(2(1) + (n-1)3)
Multiplying both sides by 2 and simplifying:
1180 = n(2 + 3n - 3)
1180 = 3n^2 - n
3n^2 - n - 1180 = 0
We can solve this quadratic equation using the quadratic formula:
n = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 3, b = -1, and c = -1180. Plugging these values in:
n = (1 ± sqrt(1 - 4(3)(-1180))) / 2(3)
n = (1 ± sqrt(1 + 14160)) / 6
n = (1 ± sqrt(14161)) / 6
n ≈ 22.98 or n ≈ -5.98
Since n must be a positive integer, we can reject the negative solution. Therefore, we have:
n ≈ 23
This means that the sum has 23 terms. To find x, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the number of terms.
Substituting the known values, we have:
x = 1 + (23-1)3
x = 1 + 22*3
x = 67
Therefore, the value of x is 67.
If the numbers a b c d e form ap then find the value of a-4b+bc-4d+e
If the given numbers a, b, c, d, e form an AP, we can express them as:
a = p
b = p + q
c = p + 2q
d = p + 3q
e = p + 4q
where p is the first term and q is the common difference.
Substituting these values, we have:
a - 4b + bc - 4d + e = p - 4(p + q) + (p + q)(p + 2q) - 4(p + 3q) + (p + 4q)
Simplifying this expression, we get:
a - 4b + bc - 4d + e = p + pq - 4p - 4q + p^2 + 3pq - 4p - 12q + p + 4q
Collecting like terms, we get:
a - 4b + bc - 4d + e = p^2 + 2pq - 12q
Since a, b, c, d, e form an AP, we know that the common difference q is equal to c - b or d - c or e - d. We can use this fact to eliminate q from the expression:
q = c - b = d - c = e - d
Substituting the first expression into the second and third expressions, we get:
d - c = c - b
e - d = d - c
Adding these two equations, we get:
e - b = 2(d - c)
Substituting this expression for 2(d-c) into the expression for q, we get:
q = (e - b)/2
Substituting this expression for q into the earlier expression for a-4b+bc-4d+e, we get:
a - 4b + bc - 4d + e = p^2 + (e-b) + (e-b)/2 - 12(e-b)/2
Simplifying this expression, we get:
a - 4b + bc - 4d + e = p^2 - 5(e-b)
Therefore, the value of a-4b+bc-4d+e is equal to p^2 - 5(e-b), which depends on the values of p and (e-b). Without knowing the specific values of a, b, c, d and e, we cannot determine a unique numerical answer for a-4b+bc-4d+e.
The sum of the n terms of two APs are in the ratio of (3n+8):(7n+15).find the ratio of the 12th terms?
Let a and d be the first term and common difference of the first AP, and let A and D be the first term and common difference of the second AP, respectively. Then, the sum of the first n terms of each AP can be expressed as:
S_1 = n/2(2a + (n-1)d) = n(a + (n-1)d/2)
S_2 = n/2(2A + (n-1)D) = n(A + (n-1)D/2)
We are given that the ratio of the sums of the first n terms of the two APs is (3n+8):(7n+15). Therefore, we can write:
S_1/S_2 = (3n+8)/(7n+15)
Substituting the expressions for S_1 and S_2, we get:
[n(a + (n-1)d/2)] / [n(A + (n-1)D/2)] = (3n+8)/(7n+15)
Simplifying this expression, we get:
(a + (n-1)d/2) / (A + (n-1)D/2) = (3n+8)/(7n+15)
Cross-multiplying, we get:
(7n+15)(a + (n-1)d/2) = (3n+8)(A + (n-1)D/2)
Expanding both sides and simplifying, we get:
(11a - 5A) + (21n - 11)(d - D)/2 = 0
Since this expression must be true for all values of n, the coefficients of (d-D) and 1 must both be zero. Therefore, we have:
11a - 5A = 0, and
21n - 11 = 0
Solving for a and A in terms of d and D using the first equation, we get:
a = 5/11 A
Substituting this expression for a into the first equation, we get:
A = 11/5 a
Therefore, a/A = 5/11, or A/a = 11/5. Since the 12th term of an AP with first term a and common difference d is a + 11d, and the 12th term of an AP with first term A and common difference D is A + 11D, we have:
(a + 11d) / (A + 11D) = (a/A) = 5/11
Multiplying both sides by 11, we get:
(11a + 121d) / (11A + 121D) = 5
Substituting the expression for A in terms of a from earlier, we get:
(11a + 121d) / (55a + 605d) = 5
Dividing both sides by 11, we get:
(a + 11d) / (5a + 55d) = 5/11
Therefore, the ratio of the 12th terms of the two APs is 5:11.