Question

# An arithmetic sequence whose first term is 2 has the property that its second, third, and seventh terms are consecutive terms of a geometric sequence. Determine all possible second terms of the arithmetic sequence.

Polynomial arithmetic
An arithmetic sequence whose first term is 2 has the property that its second, third, and seventh terms are consecutive terms of a geometric sequence. Determine all possible second terms of the arithmetic sequence.

Step 1 Given: First of an arithmetic sequence is 2 and has the property that its second, third, and seventh terms are consecutive terms of a geometric sequence. Step 2 To determine all possible second terms of an arithmetic sequence. Let, a be the first term of the arithmetic sequence. $$\Rightarrow a = 2.$$ And d be the common difference. So, the terms of the of the arithmetic sequence will be: $$a, a + d, a + 2d, a + 3d,...$$ $$\Rightarrow 2, 2 + d, 2 + 2d, 2 + 3d,.... - - - - {.: a = 2}$$ That is, to find all possible second terms of the arithmetic sequence. We need to find the value of d. Step 3 We have that second, third, and seventh terms of an arithmetic sequence are consecutive terms of a geometric sequence. $$\Rightarrow \frac{a+2d}{a+d} = \frac{a+6d}{a+2d}$$
$$\Rightarrow (2 + 2d)^2 = (2 + 6d)(2+d)$$
$$\Rightarrow 4 + 8d + 4d^2 = 4 + 2d + 12d + d^2$$
$$\Rightarrow 3d^2 = 6d$$
$$\Rightarrow d = 2$$ And second term of an arithmetic sequence is given by, $$2 + d = 2 + 2 = 4.$$ Therefore, the second of the arithmetic sequence is 4. Hence, an arithmetic sequence with the first term as 2 and with common ratio as 2 is given by 2, 4, 6, 8,.....