If the 3rd and the 9th term of an AP are 4 and – 8 respectively, then which term of this AP will be equal to zero ?
ANSWER
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Hint: This is a problem based on Arithmetic progression. To solve this we have to use the formula of the nth term of AP i.e. \[{t_n} = a + (n - 1)d\] and then we can compare both. So from there we will be able to find common differences and the first term. Now we just need to put the value of this, then we can get our required answer.
Complete step-by-step answer: As we know that we are given with the 3rd and 9th term of the AP. So, let us write an equation for that. Let the first term of this AP will be equal to a And the common difference of this AP will be equal to d. So, as we know that two consecutive terms of any AP differ by a common difference. So, if the first term of the AP is a. then the second term of this AP will be equal to a + d. And this goes on and the third term of the AP will be equal to a + 2d. So, by this way the ninth term of the AP will be equal to a + 8d, because each consecutive term increases by d. But we are given that the 3rd term is 4 and the 9th term is – 8. So, a + 2d = 4 (1) And, a + 8d = – 8 (2) Now to find the value of d. We subtract equation 1 from equation 2. We get, a + 8d – a – 2d = – 8 – 4 6d = – 12 d = – 2 So, putting the value of d in equation 1. We get, a + 2(– 2) = 4 a – 4 = 4 a = 8 Now we had to find the term of this AP which is equal to 0. Let the \[{n^{th}}\] term of this AP is equal to zero. As we know that \[{n^{th}}\] term of any AP is written as a + (n – 1)d. Where a is the first term and d is the common difference. As we know that the \[{n^{th}}\] term of this AP is zero. So, 0 = a + (n – 1)d (3) Now putting the value of a and d in equation 3. We get, 0 = 8 + (n – 1)(–2) 0 = 8 – 2n + 2 0 = 10 – 2n 2n = 10 So, n = 5 Hence, the 5th term of the given AP is equal to zero.
Note: Another method to find the value of a and d is we can write the equation of \[{n^{th}}\] term of the AP which is a + (n – 1)d and then equate this 4 when n = 4 and equate this equation with – 8 when n = 9. After solving these two equations we will get the value of a and d. And then we write the equation for the \[{n^{th}}\] term of the AP and equate that with 0 after putting the value of a and d. Then we will get the required value of n. This will be the easiest and efficient way to find the solution of the problem.