the first term and fourth term of a geomtric series are 256 and 2048 respctively
i) what is the value of the common ratios
ii) given the sum of the first n terms is 261888 find the value of n,
Use the formula you have for geometric series. From that you get:
1) The common ratio is 2.
2) And n = 10 terms.
+37084 256 256x 256x^2 256x^3
We know the fourth term is 2048 (given)
256x^3 = 2048 (where 'x' is the common ratio)
x^3 = 8
x=2
+37084 Sn=a1(1−r^n) / 1−r r≠1
261888 = 256 (1-2^n) / (1-2)
-261888 = 256 - 256(2^n)
262144 = 256(2^n)
1024 = 2^n
log(1024) = n log(2)
n = log(1024) / log(2) = 10
+129840 the first term and fourth term of a geomtric series are 256 and 2048 respctively
i) what is the value of the common ratios
ii) given the sum of the first n terms is 261888 find the value of n,
1) We have
2048 = 256r3 divide both sides by 256
2048/256 = r3
8 = r3 take the cube root of both sides
81/3 = 2 = r
2) We can use the sum formula for a geometric series here :
Sum = [First term ] * [ 1 - common ratio^term number ] / [ 1 - common ratio ]
261888 = 256 [ 1 - 2n ] / [1 - 2] divide both isdes by 256
1023 = [1 - 2n ] / -1
1023 = 2n - 1 add 1 to both sides
1024 = 2n take the log of both sides
log 1024 = log 2n and we can write
log 1024 = n log 2 divide both sides by log 2
log 1024 / log 2 = n
10 = n
