The sum of 16 term of AP is -504 while the sum of it 9term is -126, find the sum of it 20 term
Let's denote the first term of the arithmetic progression as 'a' and the common difference as 'd'.
The sum of the first 16 terms can be expressed as:
16/2 * [2a + (16-1)d] = -504
8 * [2a + 15d] = -504
2a + 15d = -63
Similarly, the sum of the first 9 terms can be expressed as:
9/2 * [2a + (9-1)d] = -126
9/2 * [2a + 8d] = -126
2a + 8d = -28
Now, we can solve these two equations simultaneously to find the values of 'a' and 'd'.
Multiplying the second equation by 15 and subtracting it from the first equation:
(2a + 15d) - 15(2a + 8d) = -63 - 15(-28)
2a + 15d - 30a - 120d = -63 + 420
-28a - 105d = 357
Simplifying this equation, we have:
28a + 105d = -357
Now, we can solve this new equation along with the second equation:
28a + 105d = -357 ...(1)
2a + 8d = -28 ...(2)
Multiplying equation (2) by 14 and subtracting it from equation (1):
28a + 105d - 28(2a + 8d) = -357 - 14(-28)
28a + 105d - 56a - 224d = -357 + 392
-28a - 105d - 224d = 35
-28a - 329d = 35
Now, we need to solve the new equation along with the first equation:
-28a - 329d = 35 ...(3)
2a + 15d = -63 ...(4)
Multiplying equation (4) by 16 and subtracting it from equation (3):
-28a - 329d - 16(2a + 15d) = 35 - 16(-63)
-28a - 329d - 32a - 240d = 35 + 1008
-60a - 569d = 1043
Now, we can solve this equation for 'a' and 'd'.
We have a system of two equations:
-28a - 329d = 35
-60a - 569d = 1043
Multiplying the first equation by 60 and subtracting it from the second equation:
-60a - 569d - 60(-28a - 329d) = 1043 - 60(35)
-60a - 569d + 1680a + 19740d = 1043 - 2100
1620a - 373d = -1057
Solving this equation, we get:
1620a - 373d = -1057
28a + 105d = -357
Multiplying the second equation by 1620 and the first equation by 28, we can eliminate the variable 'a':
448a + 1680d = -4996
448a - 10444d = -29696
Subtracting the second equation from the first equation:
173124d = 24700
d = 24700/173124
d = 0.1426 (approx)
Substituting the value of 'd' back into equation (2), we can find 'a':
2a + 8(0.1426) = -28
2a + 1.1408 = -28
2a = -29.1408
a = -29.1408/2
a = -14.5704 (approx)
Now that we have found the values of 'a' and 'd', we can determine the sum of the first 20 terms:
20/2 * [2(-14.5704) + (20-1)(0.1426)]
= 10 * [-29.1408 + 2.852]
= 10 * [-26.2888]
= -262.888
Therefore, the sum of the first 20 terms of the arithmetic progression is approximately -262.888.