The numbers a1, a2, a3, ldots . . form an arithmetic sequence with a1 ≠ a2. The threenumbers a1, a2 and a6 form a geometric sequence in that order. Then the commondifference of the arithmetic sequence is

Question

The numbers a1, a2, a3, ldots . . form an arithmetic sequence with a1 ≠ a2. The threenumbers a1, a2 and a6 form a geometric sequence in that order. Then the commondifference of the arithmetic sequence is (A) a1 (B) 2 a1 (C) 3 a1 (D) 4 a1

Tardigrade Admin · Accepted Answer

az2=a1 a6=a1(aa+5 d) (a1+d)2=a12+5 a1 d 2 a1 d+d2=5 a1 d 3 a1 d-d2=0 d[3 a1-d]=0 a ≠ 0 ⇒ d=3 a1

Q. The numbers $a_{1}, a_{2}, a_{3}, \dots ..$ form an arithmetic sequence with $a_{1} \neq = a_{2}$ . The threenumbers $a_{1}, a_{2}$ and $a_{6}$ form a geometric sequence in that order. Then the commondifference of the arithmetic sequence is

$a_{1}$

$2 a_{1}$

$3 a_{1}$

$4 a_{1}$

$5 a_{1}$

$a_{z}^{2} = a_{1} a_{6} = a_{1} (a_{a} + 5 d)$
$(a_{1} + d)^{2} = a_{1}^{2} + 5 a_{1} d$
$2 a_{1} d + d^{2} = 5 a_{1} d$
$3 a_{1} d - d^{2} = 0$
$d [3 a_{1} - d] = 0$
$a \neq = 0 \Rightarrow d = 3 a_{1}$

Q. The numbers a1​,a2​,a3​,….. form an arithmetic sequence with a1​=a2​. The threenumbers a1​,a2​ and a6​ form a geometric sequence in that order. Then the commondifference of the arithmetic sequence is

a1​

2a1​

3a1​

4a1​

5a1​

az2​=a1​a6​=a1​(aa​+5d) (a1​+d)2=a12​+5a1​d 2a1​d+d2=5a1​d 3a1​d−d2=0 d[3a1​−d]=0 a=0⇒d=3a1​