+148 1.) A geometric series b1 + b2 + b3 + .... + b10 has a sum of 180. Assuming that the common ratio of that series is 7/4, find the sum of the series b2 + b4 + b6+ b8 + b10.
2.) An arithmetic sequence with first term 1 and common difference not equal to 0 has second, tenth, and thirty-fourth terms that form a geometric sequence.
The fourth term in the geometric sequence agove appears as the nth term in the arithmetic sequence. Find the value of n.
1) - Sum =F x [1 - R^N] / [1 - R]
180 =F x [1 - 7/4^10] / [1 - 7/4], solve for F = First term
F =47185920 / 93808891
b2 + b4 + b6 + b8 + b10 are as follows:
SUM = (82575360 / 93808891, 252887040/93808891, 774466560/93808891, 2371803840/93808891, 7263649260/93808891) = 1260 / 11
Note: Edited for a minor error.
1) - Sum =F x [1 - R^N] / [1 - R]
180 =F x [1 - 7/4^10] / [1 - 7/4], solve for F = First term
F =47185920 / 93808891
b2 + b4 + b6 + b8 + b10 are as follows:
SUM = (82575360 / 93808891, 252887040/93808891, 774466560/93808891, 2371803840/93808891, 7263649260/93808891) = 1260 / 11
Note: Edited for a minor error.
+1252 2) We see that tenth is 10, second is 2, and thirty-fourth is 34, and \(10-2=8\) as well as \(34-10=24\).
Now, we see that \(\frac{24}{8}=3\), so the common ratio is 3.
Therefore, the fourth term in the geometric sequence is in the \(24\times 3=72\)nd term in the arithmetic sequence.
You are very welcome!
:P
2) The arithmetic sequence begins with 1 and a common difference of 0.2422526225 as follows:
(1, 1.24225 2622, 1.48450 5245, 1.72675 7867, 1.96901 049, 2.21126 3113, 2.45351 5735, 2.69576 8357, 2.93802 098, 3.18027 3602, 3.42252 6225, 3.66477 8848, 3.90703 147, 4.14928 4093, 4.39153 6715, 4.63378 9337, 4.87604 196, 5.11829 4583, 5.36054 7205, 5.60279 9828, 5.84505 245, 6.08730 5072, 6.32955 7695, 6.57181 0317, 6.81406 294, 7.05631 5563, 7.29856 8185, 7.54082 0807, 7.78307 343, 8.02532 6052, 8.26757 8675, 8.50983 1298, 8.75208 392, 8.99433 6543, 9.23658 9165, 9.47884 1787, 9.72109 441, 9.96334 7033, 10.20559 965, 10.44785 228, 10.69010 49)
The 4th term in the 1st. Geometric Sequence appears as the 8th term in this arithmetic sequence.
