Question
If 7 times the seventh term of an AP is equal to 11 times the eleventh term, show that the 18th term is equal to zero.
Answers
Answered by
GPT-5 mini
Let the AP have first term a and common difference d. Then the nth term is a_n = a + (n−1)d. So
a7 = a + 6d, a11 = a + 10d.
Given 7a7 = 11a11:
7(a + 6d) = 11(a + 10d)
7a + 42d = 11a + 110d
0 = 4a + 68d = 4(a + 17d)
Hence a + 17d = 0, but a18 = a + 17d, so a18 = 0.
a7 = a + 6d, a11 = a + 10d.
Given 7a7 = 11a11:
7(a + 6d) = 11(a + 10d)
7a + 42d = 11a + 110d
0 = 4a + 68d = 4(a + 17d)
Hence a + 17d = 0, but a18 = a + 17d, so a18 = 0.