The nth term of a sequence is n(n+1).
1. Is 51 a term in the sequence?
2. Which term is 182?
1. are there two consecutive integers that multiply to 51?
2. which two consecutive integers multiply to 182?
To determine if a number is a term in the sequence, we can substitute the value into the expression for the nth term.
1. To check if 51 is a term in the sequence, we can substitute n = 51 into the expression n(n+1).
n(n+1) = 51(51+1) = 51(52) = 2652
Since 51(52) is equal to 2652, 51 is a term in the sequence.
2. To find the term corresponding to 182, we can solve the equation n(n+1) = 182.
n(n+1) = 182
n^2 + n - 182 = 0
Factoring, we have (n-13)(n+14) = 0
Setting each factor equal to zero, we get n-13 = 0 or n+14 = 0
Solving each equation gives us n = 13 or n = -14
Since we're looking for a positive term, n = 13.
Therefore, the term corresponding to 182 is the 13th term.
To determine if a given number is a term in a sequence or find the corresponding term for a given value, we need to substitute the given number into the formula for the nth term of the sequence.
1. Is 51 a term in the sequence?
To check if 51 is a term in the sequence, we substitute it into the formula n(n+1). Let's solve for n:
n(n+1) = 51
Expanding the equation gives us:
n^2 + n = 51
Rearranging the equation:
n^2 + n - 51 = 0
Now we have a quadratic equation. We can solve this equation using factoring, completing the square, or using the quadratic formula. Let's solve it using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 1, and c = -51.
n = (-1 ± √(1^2 - 4(1)(-51))) / (2(1))
Simplifying the equation:
n = (-1 ± √(1 + 204)) / 2
n = (-1 ± √205) / 2
Since n represents the position of the term in the sequence, it should be a positive integer. By evaluating the expression for n, we find that n is not an integer. Therefore, 51 is not a term in the sequence.
2. Which term is 182?
Similarly, to find which term corresponds to the value 182 in the sequence, we substitute it into the formula n(n+1). Let's solve for n:
n(n+1) = 182
Expanding the equation yields:
n^2 + n = 182
Rearranging the equation:
n^2 + n - 182 = 0
Again, we have a quadratic equation. Using factoring, completing the square, or the quadratic formula, we can find the values of n. In this case, we will use factoring to solve the equation:
(n + 14)(n - 13) = 0
From this factorization, we get two possible solutions:
n + 14 = 0 --> n = -14
n - 13 = 0 --> n = 13
Since n represents the position of the term in the sequence, it should be a positive integer. Therefore, the term numbered 182 in the sequence is the 13th term.
The nth term of a sequence is n(n+1).
1. Is 51 a term in the sequence?
Ans: NO
because n(n+1) n=7
then its 7^2+7= 56
2. Which term is 182?
Ans: 13
n(n+1)
n^2=n
13^2+13= 182