The sum of 16 term of AP is -504 while the sum of it 9term is -126, find the sum of it 20 term

Answer 1

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.