Verified Answer :

  • The sequence shows a consistent difference of \(8\) between consecutive terms (\(-77 - (-85) = 8\), \(-69 - (-77) = 8\), etc.), identifying it as an arithmetic progression.
  • By adding the common difference \(d = 8\) to the fourth term \(a_4 = -61\), we determine the missing term \(a_5 = -61 + 8 = -53\).
  • We confirm this value is correct by checking that adding \(8\) to it results in the next known term, \(-45\).
  • The missing number in the sequence is \(\boxed{-53}\).

Explanation

  1. Analyzing the Sequence

The given sequence is \(-85, -77, -69, -61, \square, -45, -37\). Let the sequence be denoted as \(\{a_n\}\) for \(n = 1, 2, \dots, 7\). The known terms are \(a_1 = -85\), \(a_2 = -77\), \(a_3 = -69\), \(a_4 = -61\), \(a_6 = -45\), and \(a_7 = -37\). We need to find the missing term \(a_5\).

  1. Confirming the Arithmetic Progression

To identify the pattern, we examine the differences between consecutive terms:
\(a_2 - a_1 = -77 - (-85) = 8\)
\(a_3 - a_2 = -69 - (-77) = 8\)
\(a_4 - a_3 = -61 - (-69) = 8\)
Since the difference between consecutive terms is constant, the sequence is an arithmetic progression with a common difference \(d = 8\).

  1. Calculating the Missing Term

With the common difference \(d = 8\) and the fourth term \(a_4 = -61\), we can find the fifth term \(a_5\) by adding the common difference to \(a_4\):
\(a_5 = a_4 + d = -61 + 8 = -53\).

  1. Verification and Final Answer

To ensure our result is correct, we verify that the calculated term \(a_5 = -53\) maintains the pattern for the remaining sequence:
\(a_6 = a_5 + 8 = -53 + 8 = -45\), which matches the given term \(a_6 = -45\).
\(a_7 = a_6 + 8 = -45 + 8 = -37\), which matches the given term \(a_7 = -37\).
Since all conditions are satisfied, the missing number is \(-53\). The final answer is \(\boxed{-53}\).

Examples

Arithmetic sequences are fundamental in modeling predictable patterns in real life, such as savings plans where you deposit a fixed amount of money every month. For example, if you start with a balance of \(\$100\) and add \(\$50\) every month, your balance follows an arithmetic sequence: \(\$100, \$150, \$200, \dots\). By recognizing these patterns, you can easily calculate your future savings or identify missing entries in financial statements.

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