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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. - CBSE Class 10 - Mathematics

Question

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.

Solution

Here, let us take the first term of the A.P. as a and the common difference as d

We are given that 10 times the 10th term is equal to 15 times the 15th term. We need to show that 25th term is zero.

So, let us first find the two terms.

So, as we know,

a_n = a + (n - 1)d

For 10 th term (n = 10)

a_10 = a + (10 - 1)d

= a + 9d

For 15 th term (n = 15)

a_15 = a + (15 - 1)d

= a + 14d

Now, we are given,

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

Solving this we get

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

Solving this we get

10a + 90d = 15a + 210d

90d - 210d = 15a - 10a

-120d = 5a

-24d = a .......(1)

Next, we need to prove that the 25th term of the A.P. is zero. For that, let us find the 25th term using n = 25,

a_25 = a + (25 - 1)d

= -24d + 24d   (using 1)

= 0

Thus the 25 the term of the given A.P is zero

Hence proved

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Solution 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. Concept: Arithmetic Progression.
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