Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem10

Video Transcription

Question number 10. The numbers the square root of 7, the cube root of 7, and the sixth root of 7 are the first three elements of a geometric sequence. The  fourth element of the sequence is which of the following? We recall that the sequence is geometric provided that there is a common ratio between elements.  And so what I mean by that is if we have a sequence beginning with A1, A2, A3, A4, like that, then here A2 is some number r, the common ratio, times the previous  element, A3 is some number r times the previous element A2, and so on. So we are looking for this common ratio r over here, and we can solve for r. r is equal  to A2 divided by A1, and in our example, we have A1 is equal to the square root of 7, A2 is equal to the cube root of 7, so we calculate r as A2, the cube root  of 7 divided by the square root of 7, and that is 7 to the power one-third divided by 7 to the power one-half. Subtracting exponents, we have 7 to the  power one-third minus one-half, and that comes out to 7 to the power negative 1 over 6. So that's the common ratio, and then we know that A4, the element we're  looking for, is r times A3, which is 7 to the negative one-sixth times A3, and A3 here is the sixth root of 7. So we have 7 to the negative one-sixth times 7 to the  positive one-sixth. The exponents add to 0, and 7 to the 0 power is another name for 1. So we have found our answer here. The next element, the fourth element in  that geometric sequence with common ratio of 7 to the power negative one-sixth is 1, so the answer here should be E.

Video Summary

Summary Not Available