The nineth term of an AP is -32, and the sum of eleventh and thirteenth terms is -94. Find the common difference of the AP.?

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Answer

Problem: The nineth term of an AP is -32, and the sum of eleventh and thirteenth terms is -94. Find the common difference of the AP.

Solution:
Let the first term of the AP be 'a', and the common difference be 'd'.

Step 1: Find the ninth term of the AP
The ninth term of the AP can be found using the formula:
a + (n-1)d, where n is the term number.
So, the ninth term can be written as:
a + 8d = -32 ...(1)

Step 2: Find the sum of eleventh and thirteenth terms
The sum of two terms of an AP can be found using the formula:
2a + (n-2)d, where n is the term number of the second term.
So, the sum of eleventh and thirteenth terms can be written as:
2a + 20d + 2d = -94
2a + 22d = -94 ...(2)

Step 3: Solve the equations (1) and (2) simultaneously
We can solve the equations (1) and (2) simultaneously to find the values of 'a' and 'd'.
Multiplying equation (1) by 11 and equation (2) by 4, we get:
11a + 88d = -352 ...(3)
8a + 88d = -376 ...(4)
Subtracting equation (4) from equation (3), we get:
3a = 24
a = 8

Substituting the value of 'a' in equation (1), we get:
8 + 8d = -32
8d = -40
d = -5

Step 4: Verify the solution
We can verify the solution by checking if the ninth term is -32 and the sum of eleventh and thirteenth terms is -94.
The ninth term can be found using the formula: a + (n-1)d
Substituting the values of 'a' and 'd', we get:
a9 = 8 + (9-1)(-5) = -32
So, the ninth term is -32.

The sum of eleventh and thirteenth terms can be found using the formula: 2a + (n-2)d
Substituting the values of 'a' and 'd', we get:
a11 + a13 = 2a + (11+13-2)(-5) = -94
So, the sum of eleventh and thirteenth terms is -94.

Therefore, the common difference of the AP is -5.

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