In AP with a small number of terms, you can use the following format for ease in calculations.
1) Where the number of terms is odd, take the middle term as x and common difference as y.
- For a 3-term AP, use the terms (x – y), x, (x + y)
- For a 5-term AP, use the terms (x – 2y), (x – y), x, (x + y), (x + 2y)
2) Where the number of terms is even, take the middle terms as x – y and x + y and common difference as 2y.
- For a 4-term AP, use the terms (x – 3y), (x – y), (x + y), (x + 3y)
- For a 6-term AP, use the terms (x – 5y), (x – 3y), (x – y), (x + y), (x + 3y), (x + 5y)
Example 9
An AP has 3 terms. If the sum of terms is 24 and the product of terms is 440. Then, what is the third term in this A.P ?
(1) 5
(2) 11
(3) 5or11 (4) None of the above
Solution
Let the 3 terms be (x – y), x, (x + y).
Sum of 3 terms = 3x = 24 ⇒ x=8
Product of 3 terms = (x−y)×x×(x+y)=440
⇒ (8−y)×8×(8+y)=440
⇒ 64−y2=55
⇒ y2=9
⇒ y = ±3
First term =(8+3)or(8–3) = 11 or 5
Answer:(3)5or11
Example 10
If the sum of 6 terms of an AP is 69 and the 5thterm is 16, then what is the 3th term?
Solution
Let the 6 terms be (x – 5y), (x – 3y), (x – y), (x + y), (x + 3y), (x + 5y)
Sum of 6 terms = 6x=69⇒x=11.5
5th term = x+3y=16
⇒ 11.5+3y=16
⇒ y=1.5
∴3th term =x−y = 11.5 – 1.5 = 10
Answer: 10
2.4 Inserting Arithmetic Means
When Arithmetic Means are inserted between two numbers, say x and y , then all these numbers together would form an Arithmetic Progression where x and y will be the first and last terms respectively.
In any AP with n terms, there are (n – 2) arithmetic means between the first and the last terms. No direct formula is required for this. Please logically apply the AP formulae where required.
Example 11
If 20 Arithmetic Means are inserted between 14 and 16, then what is the sum of these Arithmetic Means?
Solution
Note that the 20 Arithmetic Means along with 14 and 16 form an AP.
Average of this AP = 214+16=15
Average of the AP with just the 20 AMs will also be 15.
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