Advertisements
Advertisements
प्रश्न
If a, b, c are in geometric progression, and if `"a"^(1/x) = "b"^(1/y) = "C"^(1/z)`, then prove that x, y, z are in arithmetic progression
उत्तर
Given a, b, c are in G.P.
⇒ b2 = ac
⇒ log b2 = log ac
(i.e.) 2log b = log a + log c .....(1)
We are given `"a"^(1/x) = "b"^(1/y) = "C"^(1/z)` = k ......(say)
⇒ log ak = `1/x`
log bk = `1/y`
log ck = `1/z`
⇒ log ak = `1/x`
⇒ x = log ka
y = log kb
z = log kc
Substituting these values in equation (1) we get
2y = x + z
⇒ x, y z are in A.P.
APPEARS IN
संबंधित प्रश्न
Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them
`(("n" + 1)("n" + 2))/(("n" + 3)("n" + 4))`
Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them
`4 (1/2)^"n"`
Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them
`(2"n" + 3)/(3"n" + 4)`
Write the first 6 terms of the sequences whose nth term an is given below
an = `{{:(1, "if n" = 1),(2, "if n" = 2),("a"_("n" - 1) + "a"_("n" - 2), "if n" > 2):}}`
Write the first 6 terms of the sequences whose nth term an is given below
an = `{{:("n", "if n is" 1"," 2 "or" 3),("a"^("n" - 1) + "a"_("n" - 2) + "a"_("n" - 3), "if n" > 3):}`
Write the nth term of the following sequences.
2, 2, 4, 4, 6, 6, . . .
Write the nth term of the following sequences.
`1/2, 2/3, 3/4, 4/5, 5/6, ...`
Write the nth term of the following sequences.
6, 10, 4, 12, 2, 14, 0, 16, −2, . . .
The product of three increasing numbers in GP is 5832. If we add 6 to the second number and 9 to the third number, then resulting numbers form an AP. Find the numbers in GP
Write the nth term of the sequence `3/(1^2 2^2), 5/(2^2 3^2), 7/(3^2 4^2), ...` as a difference of two terms
If tk is the kth term of a G.P., then show that tn – k, tn, tn + k also form a GP for any positive integer k
If the roots of the equation (q – r)x2 + (r – p)x + p – q = 0 are equal, then show that p, q and r are in AP
If a , b , c are respectively the pth, qth and rth terms of a G . P show that (q – r) log a + (r – p) log b + (p – q) log c = 0
Choose the correct alternative:
If a, 8, b are in A.P, a, 4, b are in G.P, if a, x, b are in HP then x is
Choose the correct alternative:
The sequence = `1/sqrt(3), 1/(sqrt(3) + sqrt(2)), 1/(sqrt(3) + 2sqrt(2)) ...` form an
Choose the correct alternative:
The nth term of the sequence 1, 2, 4, 7, 11, …… is
Choose the correct alternative:
The nth term of the sequence `1/2, 3/4, 7/8, 15/16, ...` is
