 Hello friends and how are you all doing today? The question says, if 5 times the fifth term of an AP is equal to 8 times its eighth term, show that its thirteenth term is zero. Let us first write down the fifth term of an AP. Let the fifth term of an AP, that is the arithmetic progression be equal to a5, that is equal to a plus 4d, where a is the first number of the arithmetic progression and d is the difference between the two consecutive numbers of an AP. Similarly the eighth term of an AP is equal to a plus 7d. Now we will be writing down the equation according to the question. It is given to us 5 times the fifth term is equal to 8 times the eighth term. Then we need to show that thirteenth term is zero. So let us open the brackets. We have 5a plus 20d is equal to 8a plus 56. Further we have 8a minus 5a is equal to, we have d with it too, equal to 20d minus 56d which further implies 3a is equal to minus 36d. On dividing the both sides by 3 we have 3a upon 3 equal to minus 36 upon 3d. Further we have a equal to minus 12d which further implies a plus 12d is equal to zero. Now we know that a thirteenth term of an AP is written as a plus 12d. So this shows that thirteenth term of an AP where the first term is a and the difference between two consecutive numbers of an AP is d is equal to zero. Right? So this completes the session. Hope you understood it well and enjoyed it too. Have a nice day.