# If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

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Updated on: September 25th, 2023

If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, it is proved that the 25th term of the AP is zero. It is given that:

10 x 10th term =15 x 15th term

Let d be a common difference and a be the first term.

10 [a + (n – 1)d] = 15 [a + (n – 1)d]

On simplifying:

10 (a + 9d) = 15 (a + 14d)

5a + 120d = 0

a + 24d = 0 …. (i)

Now,

25th term

= a + (25 – 1)d

= a + 24d

= 0

Hence, 25th term=0

### Notation in Arithmetic Progression

It is referred to as AP, a mathematical sequence in which there is always a constant difference between two terms. The following are some of the key terms we will encounter in AP:

1. First term (a)
2. Common difference (d)
3. nth Term (an)
4. Sum of the first n terms (Sn)

### Types of Arithmetic Progression

Finite AP: An AP is referred to as finite AP if it has a finite number of terms. The last term exists in a finite AP.

For example 3,5,7,9,11,13,15,17,19,21

An AP that has an unlimited number of terms is referred to as an infinite AP. Such APs lack a final term.

For example: 5,10,15,20,25,30, 35,40,45………………

Summary:

## If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

It is demonstrated that the 25th term of an AP is zero if 10 times the 10th term of an AP equals 15 times the 15th term. An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.

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