how many terms of the series 54+51+48+45......must be taken to make 513?explain the double answer?

Answer 1

if you have this arithmetic progresion
a,a+r,a+2r............a+(n-1)r the sum is n*a+r*n*(n-1)/2
so in this case
54 *n -3*(n^2-n)/2>= 513
108 n -3n^2 +3n>= 1026
3n^2 -111n+1056=0 solving the 2nd degreeequ.
n =(111+-3)6= 18 or 19. but ther3 are only 18 terms so the solution is 18

Answer 2

This is an arithmetic progression with common difference
d= -3 and first term a = 54. The sum S to n terms of an AP is
S=n/2*{2a+(n-1)*d} Substituting, we get
513=n/2*{2*54+(n-1)*(-3)} Multiplying both sides by 2 and simplifying, we get 1026=n{108-3n+3}=n(111-3n)
Hence, 1026=-3n^2+111n or 3n^2-111n+1026=0.
Solving this quadratic, we get n=18 or 19.
The 19th term in the given series would be 54+{(19-1)(-3) or 0. Hence whether we take n=18 or n=19, the answer would be the same for the given series totaling 513, since the 19th term is 0.
Thus the no. of terms required could be 18 or 19, and both would give the same sum of 513.
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An alternative solution would be to divide the series by 3 and the question would then be to find n when the first term is 18, common difference -1 and the sum to n terms 171. The series would be 18,17,16,15,.,,3,2,1 (18 terms) the total of which would be 171. Hence taking also the next term in the series as 19th term which is of value 0, the total would still be 171.

Answer 3

This is an arithmetic series with a = 54, d = -3.

The sum to n terms is
(n/2)(2a + (n-1)d)
= (n/2)(108-3(n-1))
= (n/2)(111-3n)

For this to be 513, we need
n((111-3n) = 1026
3n(37-n) = 1026
n(37-n) = 342
n^2 - 37n + 342 = 0
2*3*3*19 = 342
(n-18)(n-19) = 0
Hence the sum of 513is obtained from 18 terms and from 19 terms.

The reason for this is that the 19th term is 0.
T(n) = 54 - 3(n-1) and so
T(19) = 54 - 3*18 = 0.

The sum increases for the first 18 terms, but all terms after the 19th are negative, and so the sums are decreasing after that.

Answer 4

Sn = a + (a + d) + (a + 2d) + (a + 3-d) + ---[a + (n - a million)d] Sn = [a + (n - a million) d] + [a + (n - 2) d] + ------a 2 Sn = n [ 2a + (n - a million) d ] Sn = (n/2) [ 2a + (n - a million) d ] 513 = n/2 [ 108 + ( n - a million ) (-3) ] 1026 = 108 n + (-3) (n) (n - a million) 1026 = 108 n - 3 n ² + 3n 3 n ² - 111 n + 1026 = 0 n ² - 37 n + 342 = 0 n(n-19)-18(n-19)=0 (n-19)(n-18)=0 for this reason n = 19 or 18

Answer 5

54+51+48+45+42+39+36+33+30+27+
24+21+18+15+12+9+6+3 = 513

You need 18 terms. I'm not sure on the double answer part sorry.
Maybe it's because you need to work out the series is reducing by 3 so that you can complete the question on how many terms you need.
I'm not doing your homework am I?!

Answer 6

First term = 54

Difference = -3

So, n(2*54+(n-1)*-3) = 2*513
or 111n-3n^2 = 1026
or n^2-37n+342 = 0
or n =18 or 19

The double answer arises because the 19th term is 0.

Answer 7

18 terms

n, n - 3 , n - 6 , n-9 , n-12.................... n - 51

18n - 459 = 513

n=54

Answer 8

18 & 19?

Choose whether or not to +0????