Question

6.If 5 times the 5^th term of an AP is equal to 10 times the 10^ th term show that its 15^th term is zero​

Answer 1

Step-by-step explanation:

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

$\longmapsto \bf a_n=a+(n-1)d$

where

a is the first term

d is the common difference

5th term :

a₅ = a + (5 - 1)d

a₅ = a + 4d

10th term :

a₁₀ = a + (10 - 1)d

a₁₀ = a + 9d

Given,

5 × 5th term = 10 × 10th term

5(a + 4d) = 10(a + 9d)

5a + 20d = 10a + 90d

 5a - 10a = 90d - 20d

  -5a = 70d

  a = 70d/-5

   a = -14d

   a + 14d = 0  ➟ eqn. 1

15th term :

a₁₅ = a + (15 - 1)d

a₁₅ = a + 14d

a₁₅ = 0 [∵ eqn. 1]

Hence, 15th term is zero.

Answer 2

Question:-

If 5 times the 5^th term of an AP is equal to 10 times the 10^ th term show that its 15^th term is zero.

Required Answer:-

Given:-

• 5 times the 5^th term of an AP is equal to 10 times the 10^th term.

To Do:-

• Show that 15^th term = 0

Solution:-

We know that,

• n^th term of an arithmetic progression (AP) = a+(n−1)d

Where,

• a is the first term of AP
• d is common difference
• n is number of terms

We were given that:-

• The 5 times the 5^th term is equal to 10 times the 10^th term.

Therefore,

5(5^th term) = 10(10^th term)

⟹ 5{a+(5−1)d} = 10{a+(10−1)d}

⟹ 5(a+4d) = 10(a+9d)

⟹5a+20d = 10a+90d

⟹ 5a−10a = 90d−20d

⟹ -5a = 70d

⟹ a = −14d

Now,

15^th term of the AP

= a+(n−1)d

= a + (15-1)d

= -14d + 14d

= 0

Hence, 15^th term of the AP = 0

Proved!