13. If 6 times the 6th term of an A. P. is equal to 13 times the 13th term, prove that 19th term
of this A. P. is zero.

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see explanation

Step-by-step explanation:

The n th term of an AP is

= a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Given

6(a₁ + 5d) = 13(a₁ + 12d) ← distribute parenthesis on both sides

6a₁ + 30d = 13a₁ + 156d ( subtract 13a₁ from both sides )

- 7a₁ + 30d = 156d ( subtract 30d from both sides )

- 7a₁ = 126d ( divide both sides by - 7 )

a₁ = - 18d

Now

a₁₉ = a₁ + 18d = - 18d + 18d = 0 ← as required

2500

step-by-step explanation: