The sum of the 2nd and 3rd terms of an arithmetic progression (AP) is 21. The product of the 4th and 5th terms is 414. Find the 9th term of the AP.
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The sum of the 2nd and 3rd terms of an arithmetic progression (AP) is 21. The product of the 4th and 5th terms is 414. Find the 9th term of the AP.
Let’s denote the first term of the arithmetic progression (AP) as ( a ) and the common difference as ( d ).
1. The second term is ( a + d ).
2. The third term is ( a + 2d ).
The sum of the 2nd and 3rd terms is given as:
[
(a + d) + (a + 2d) = 21
]
This simplifies to:
[
2a + 3d = 21 quad text{(1)}
]
Next, the fourth term is ( a + 3d ) and the fifth term is ( a + 4d ).
The product of the 4th and 5th terms is given as:
[
(a + 3d)(a + 4d) = 414
]
Expanding this, we get:
[
a^2 + 7ad + 12d^2 = 414 quad text{(2)}
]
Now we have two equations (1) and (2). We will solve them step by step.
From equation (1):
[
2a + 3d = 21 implies a = frac{21 – 3d}{2}
]
Plugging ( a ) from this into equation (2):
[
left(frac{21 – 3d}{2}right)^2 + 7