Question

sequences
find the following for the given sequence
78, 68, 58, 48…
explicit formula
20th term
set up the formula to find the sum of 20 terms of the sequence.
find the sum of the 20 terms of the sequence

Explanation:

Step1: Identify Sequence Type

The sequence is 78, 68, 58, 48... Check the common difference: \(68 - 78 = -10\), \(58 - 68 = -10\), \(48 - 58 = -10\). So it's an arithmetic sequence with first term \(a_1 = 78\), common difference \(d = -10\).

Step2: Explicit Formula for Arithmetic Sequence

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = 78\) and \(d = -10\):
\(a_n = 78 + (n - 1)(-10) = 78 - 10n + 10 = 88 - 10n\).

Step3: Find 20th Term

Use the explicit formula with \(n = 20\):
\(a_{20} = 88 - 10(20) = 88 - 200 = -112\).

Step4: Sum of First \(n\) Terms of Arithmetic Sequence

The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2}(a_1 + a_n)\) or \(S_n = \frac{n}{2}[2a_1 + (n - 1)d]\). We can use either. Let's use \(S_n = \frac{n}{2}(a_1 + a_n)\). We know \(n = 20\), \(a_1 = 78\), \(a_{20} = -112\).

Step5: Calculate Sum of 20 Terms

Substitute into the sum formula:
\(S_{20} = \frac{20}{2}(78 + (-112)) = 10(78 - 112) = 10(-34) = -340\).

Answer:

s:

• Explicit Formula: \(a_n = 88 - 10n\)
• 20th Term: \(-112\)
• Sum Formula Setup: \(S_{20} = \frac{20}{2}(78 + a_{20})\) (or \(S_{20} = \frac{20}{2}[2(78) + 19(-10)]\))
• Sum of 20 Terms: \(-340\)