Consider the sequence 13;11;9;7;5...
A) show that Tn+T15-n=0
looks like an AS, with a = 13 , d = -2
tn + t(15-n)
= a + (n-1)d + a + (14-n)d
= a + nd - d + a + 14d - nd
= 2a + 13d
= 26 - 26
= 0
How did 15 change into 14
How did 15 change to 14
Oh, I see what you did there! You want to prove that the sum of the nth term and the (15-n)th term of the sequence is zero. Well, let's take a closer look.
If we examine the sequence, we can notice that each term is decreasing by 2 units. So to find the nth term, we can use the formula Tn = 13 - 2(n-1).
Now, let's prove the given equation Tn + T15-n = 0.
Substituting the formula for the nth term, we get:
(13 - 2(n-1)) + (13 - 2((15-n)-1)) = 0
Simplifying further, we have:
(13 - 2n + 2) + (13 - 2(14 - n)) = 0
(15 - 2n) + (13 - 28 + 2n) = 0
15 - 2n + 13 - 28 + 2n = 0
15 - 28 + 13 = 0
0 = 0
So, we can see that Tn + T15-n does equal zero, confirming the given equation.
Now, if you're looking for me to crack a joke about this, I'm afraid math jokes can be a bit tricky. You might say math puns are the square root of all evil!
To show that Tn + T15-n = 0, we need to find the expression for Tn and T15-n in the given sequence and prove that they add up to zero.
In the given sequence, we can see that the terms are decreasing by 2 each time.
So, the expression for the nth term (Tn) can be represented as:
Tn = 13 - (n-1) * 2
Similarly, the expression for the (15 - n)th term (T15-n) can be represented as:
T15-n = 13 - [(15 - n) - 1] * 2
Simplifying the expressions, we have:
Tn = 15 - 2n
T15-n = 15 - 2(15 - n - 1) = 15 - 2(14 - n) = 15 - 28 + 2n = -13 + 2n
Now, let's substitute these expressions into the equation Tn + T15-n = 0 and prove that it holds true.
Tn + T15-n = (15 - 2n) + (-13 + 2n)
= 15 - 2n - 13 + 2n
= 15 - 13 - 2n + 2n
= 2
As we can see, Tn + T15-n = 2, and it does not equal zero. Therefore, the equation Tn + T15-n = 0 does not hold true for the given sequence 13;11;9;7;5...
If you have any further questions, please, let me know.
a=13, d=-2
Tn+T15-n
a+(n-1)d + a+(15-n)d
a+dn-d + a+15d-dn
2a + 14d
2(13)+14(-2)
26-28= -2
* -2 < 15