Question

An arithmetic Progression is such that the tenth term is equal to the sum of the first ten terms.
Show that the fifth term must be zero.​

Answer 1

Step-by-step explanation:

Given :

In an A.P., the tenth term is equal to the sum of the first ten terms.

To prove :

the fifth term is zero

Solution :

The nth term of an A.P. is given by,

 $\underline{\boxed{\bf a_n=a+(n-1)d}}$

where

a denotes the first term

d denotes common difference

Let's find the tenth term of the A.P. :

Put n = 10,

a₁₀ = a + (10 - 1)d

a₁₀ = a + 9d

The sum of first n terms is given by,

 $\underline{\boxed{\bf S_n=\dfrac{n}{2}[2a+(n-1)d]}}$

Finding the sum of first ten terms of the A.P. :

Put n = 10

$\tt S_{10}=\dfrac{10}{2}[2a+(10-1)d]$

S₁₀ = 5[2a + 9d)

S₁₀ = 10a + 45d

As given,

a₁₀ = S₁₀

a + 9d = 10a + 45d

10a + 45d - a - 9d = 0

  9a + 36d = 0

  9(a + 4d) = 0

   a + 4d = 0/9

  a + 4d = 0

Find the fifth term of the A.P. :

a₅ = a + (5 - 1)d

a₅ = a + 4d

a₅ = 0

Hence, the fifth term must be zero so that the tenth term is equal to the sum of the first ten terms.