For other viewers, the exercise and solution are as follows:
8 . . . . .An arithmetic series has first term
a and common difference
d.
. . . . . ..The sum of the first five terms of the series is
575.
. .(a). . .Show that
a + 2d = 115.
. .(b). . .Given that the tenth term of the series is
87, find the value of
d.
. .(c). . .The
n-th term of the series is
un. Given that
uk > 0 and
uk+1 < 0, find the value of:
. . . . . . . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 1}^k \, u_n\)
8(a). . .\(\displaystyle S_5\, =\, \dfrac{5}{2}\, \bigg[\, 2a\, +\, (5\, -\, 1)d\, \bigg]\)
. . . . . .\(\displaystyle \dfrac{5}{2}\, \bigg[\, 2a\, +\, (5\, -\, 1)d\, \bigg]\, =\, 575;\, \mbox{ }\, 5\, (2a\, +\, 4d)\, =\, 575\, \times\, 2\)
. . . . . .\(\displaystyle 2a\, +\, 4d\, =\, 115\, \times\, 2\, \Rightarrow\, a\, +\, 2d\, =\, 115\)
8(b). . .\(\displaystyle a\, +\, (10\, -\, 1)d\, =\, 87\)
. . . . . .\(\displaystyle a\, +\, 2d\, =\, 115,\, a\, +\, 9d\, =\, 87\, \Rightarrow\, 7d\, =\, 87\, -\, 115\)
. . . . . .\(\displaystyle 7d\, =\, -28,\, d\, =\, -4\)
8(c). . .\(\displaystyle \mbox{When }\, d\, =\, -4,\, a\, =\, 123.\)
. . . . . .\(\displaystyle u_k\, =\, 123\, +\, (k\, -\, 1)(-4)\, >\, 0\)
. . . . . .\(\displaystyle u_{k+1}\, =\, 123\, +\, (k)(-4)\, <\, 0\)
. . . . . .\(\displaystyle k\, <\, 31.75,\, k\, >\, 30.75\, \Rightarrow\, k\, =\, 31\)
. . . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 1}^{k}\, u_n\, =\, \dfrac{31}{2}\, \bigg[\, 2a\, +\, (31\, -\, 1)d\, \bigg]\, =\, 1953\)
I've got a problem to solve a problem 8.C)
the problem is the value of n.
No; the issue is finding the value of
k, not
n.
I construct the inequalities but after i have got two value and i dont know what to pick 31.75 and 30.75.
What do you mean by "what to pick"? You were given two inequalities, you solved the two inequalities, and thus arrived at bounds on the value of k , which is the value you need to find. Where is the issue?
In the solution papers there is n value 31
No; they have
k = 31 , because
k is bounded by the in equalities you say you solved:
30.75 < k < 31.75 ,
k a whole number.
and i have no idea how they come up with it. Could anybody explain, please.
They just write 31.75 =>31 and 30.75=>31 and I just dont get it.
No, they don't. They show that
k, a whole number, is less than
31.75 and greater than
30.75. What who le number is between
30.75 and
31.75?
As for how they got
31.75 and
30.75, what did
you get when you solved the two given inequalities?
Please show all of your steps. Thank you!
