how many terms are there in the ap 9,13,17,21........97?
Answers
Answer: 12
Step-by-step explanation:
I’ll denote the final term in the progression (i.e. the one that brings the sum to 636) as x. Notice that the average value of the terms is then halfway between the first term and last term, so the average is x+92.
Next, notice that if we take any term in the sequence, subtract 1 and then divide the result by 8, the answer tells us WHICH term in the sequence it is. For example, 9−18=1 so 9 is the first term and 17−18=2 so 17 is the second term and so on. From this, we learn that x is term number x−18.
Now, if the average value of the terms is x+92 and there are x−18 terms total, we can conclude that their sum must be
x+92⋅x−18
Since this function of x must be 636, we learn that
(x+9)(x−1)=16⋅636
Rather than trying to expand this and then solving the resulting expression with the quadratic equation, let’s take a simpler approach. The left hand side of the equation is the product of two positive integers that are 10 apart. (We know they must be positive integers because all the terms in the progression differ by exactly 8.)
So all we need to do is to find two positive integers that are 10 apart that multiply to be 16⋅636. To find them, let’s write down the prime factorization of 636⋅16. I notice that six divides 636 leaving 106=2⋅53 and the work is done.
16⋅636=26⋅3⋅53
Now our goal is to rearrange these prime factors to get two numbers that differ by 10. That’s easier than you think because 53 is so large compared to 2 and 3. We notice that 53 by itself is to small to be one of the numbers (because the other would be 26⋅3=192) and 53⋅3 is too large to be one of the numbers (because the other would be 26=64), so it must be that 53⋅2=106 is one of the factors — assuming the problem actually has a solution.
If 53⋅2=106 is one of the factors. then the other must be 25⋅3=96 and indeed we found two factors that differ by 10.
We see that (x+9)(x−1)=16⋅636=106⋅96 which immediately tells us that x−1=96 and then x=97. The last term we need in our progression must be 97. Of course, you asked for how many terms we need, but that is simple enough since we already noticed that the term number of x is just x−18. Apply this result to x=97 and we see that 97 is term number 97−18=12.