Find the missing term in this geometric sequence: 2, [ ? ], 98
The correct answer and explanation is:
The geometric sequence given is: 2, [ ? ], 98.
To find the missing term, we need to understand the properties of a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio (r).
Let the missing term be xx. The sequence is then:
2,x,982, \quad x, \quad 98
The common ratio can be expressed as: r=x2=98xr = \frac{x}{2} = \frac{98}{x}
Since the ratio between consecutive terms is constant, these two ratios must be equal: x2=98x\frac{x}{2} = \frac{98}{x}
Cross-multiplying gives: x2=2×98x^2 = 2 \times 98 x2=196x^2 = 196
Taking the square root of both sides: x=±196=±14x = \pm \sqrt{196} = \pm 14
Thus, the missing term could be either 1414 or −14-14.
To verify, check if the sequence forms a geometric progression with each candidate:
- For x=14x = 14, the ratios are 142=7\frac{14}{2} = 7 and 9814=7\frac{98}{14} = 7. The common ratio is 7, so the sequence is 2,14,982, 14, 98.
- For x=−14x = -14, the ratios are −142=−7\frac{-14}{2} = -7 and 98−14=−7\frac{98}{-14} = -7. The common ratio is -7, so the sequence is 2,−14,982, -14, 98.
Both values satisfy the condition of a geometric sequence, so the missing term could be either 1414 or −14-14, depending on the context or additional constraints.
Summary:
- The missing term xx is found by setting the ratio between terms equal.
- Solving x2=98x\frac{x}{2} = \frac{98}{x} yields x2=196x^2 = 196.
- Taking the square root gives two possible values: 1414 or −14-14.
- Both values form valid geometric sequences with common ratios of 7 or -7, respectively.
Hence, the missing term in the geometric sequence is either 14 or -14.