To determine the next term of the arithmetic progression (AP) given the terms [tex]\(\sqrt{18}\)[/tex], [tex]\(\sqrt{50}\)[/tex], and [tex]\(\sqrt{98}\)[/tex], we need to analyze the sequence step-by-step.
1. Identify the given terms by squaring them:
- First term: [tex]\(\sqrt{18} \rightarrow 18\)[/tex]
- Second term: [tex]\(\sqrt{50} \rightarrow 50\)[/tex]
- Third term: [tex]\(\sqrt{98} \rightarrow 98\)[/tex]
2. Calculate the differences between consecutive terms:
- Difference between the first and the second term: [tex]\(50 - 18 = 32\)[/tex]
- Difference between the second and the third term: [tex]\(98 - 50 = 48\)[/tex]
3. Determine the common difference in the AP:
- For an arithmetic progression, the common difference should ideally be the same. Here, we can take the average of the two differences to ensure a consistent value across the sequence.
- Average common difference: [tex]\(\frac{32 + 48}{2} = \frac{80}{2} = 40\)[/tex]
4. Calculate the next term:
- The next term in the sequence can be found by adding the common difference to the last known term.
- Next term: [tex]\(98 + 40 = 138\)[/tex]
5. Find the square root of the next term:
- [tex]\(\sqrt{138} \approx 11.75\)[/tex]
6. Determine which option corresponds to the next term:
- Given options:
- [tex]\(\sqrt{146} \approx 12.08\)[/tex]
- [tex]\(\sqrt{128} \approx 11.31\)[/tex]
- [tex]\(\sqrt{162} \approx 12.73\)[/tex]
- The closest integer approximation of [tex]\(\sqrt{138}\)[/tex] is [tex]\(\sqrt{128}\)[/tex], which aligns best with the calculated value.
Therefore, the next term of the arithmetic progression is [tex]\(\sqrt{128}\)[/tex].
The correct answer is:
(b) [tex]\(\sqrt{128}\)[/tex].
