The sum of 1st five terms of an AP nd the sum of the 1st seven termd of the same AP is 167. If the sum of the 1st ten terms of this AP is 235, find the sum of its1st twenty terms
Simply stating the answer serves the student no purpose.
The 1st 20 terms = 970
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To find the sum of the first 20 terms of an arithmetic progression (AP), we need to first find the common difference (d) and the first term (a).
Let's start by finding the common difference (d) using the given information.
The sum of the first five terms of the AP is 167, which can be expressed as:
S5 = 167
The sum of the first seven terms of the AP is also given as 167:
S7 = 167
We can use these two equations to find the common difference (d).
The sum of 'n' terms in an AP can be calculated using the formula:
Sn = (n/2) * (2a + (n - 1)d)
For S5, substituting n = 5, we have:
S5 = (5/2) * (2a + (5 - 1)d) = 167 --> (1)
For S7, substituting n = 7, we have:
S7 = (7/2) * (2a + (7 - 1)d) = 167 --> (2)
Dividing equation (2) by equation (1), the terms involving 'a' will cancel out, and we can solve for 'd':
S7 / S5 = [(7/2) * (2a + 6d)] / [(5/2) * (2a + 4d)]
Simplifying the equation:
[(7/5) * (2a + 6d)] / [(2a + 4d)] = 1
Cross-multiplying:
(7/5) * (2a + 6d) = 2a + 4d
Expanding:
(14a + 42d) / 5 = 2a + 4d
Multiplying through by 5:
14a + 42d = 10a + 20d
Simplifying:
4a = -22d --> a = (-22/4) * d --> a = (-11/2) * d --> (3)
Now that we have found the relation between 'a' and 'd', we can proceed to find the values of 'a' and 'd' using the given information about the sum of the first ten terms.
The sum of the first ten terms of the AP is given as 235:
S10 = 235
Using the formula for Sn, substituting n = 10, we have:
S10 = (10/2) * (2a + (10 - 1)d) = 235
Simplifying the equation:
5 * (2a + 9d) = 235
Dividing both sides by 5:
2a + 9d = 47 --> (4)
Now that we have two equations (3) and (4) with two unknowns (a and d), we can solve them simultaneously to find the values of 'a' and 'd'.
Substituting equation (3) into equation (4):
2((-11/2)d) + 9d = 47
Simplifying:
-11d + 9d = 47
Combining like terms:
-2d = 47
Dividing by -2:
d = -47/2 = -23.5
Substituting the value of 'd' into equation (3) to find 'a':
a = (-11/2) * (-23.5) = 126.5
Now that we have found the values of 'a' and 'd' as 126.5 and -23.5, respectively, we can calculate the sum of the first 20 terms (S20) using the formula for Sn:
S20 = (20/2) * (2a + (20 - 1)d)
= 10 * (2 * 126.5 + (20 - 1) * (-23.5))
= 10 * (253 + 19 * (-23.5))
= 10 * (253 - 446.5)
= 10 * (-193.5)
= -1935
Therefore, the sum of the first 20 terms of the given AP is -1935.