A sequence of numbers, S1, S2, S3, . . ., is defined by Sk = k2 – k fo

\(S_k = k^2 – k\)

ie.. \(S_n = n^2 – n = 110\)

i.e. n*(n-1) = 110

i.e. n = 11 or -10 but since n i spositive hence

n = 11

Now, \(S_{n-1} = S_{10} = 10^2 - 10 = 90\)

Answer: Option D

Our first job is to determine which term has a value of 110.

We'll use the formula for determining the value of each term \(S_k = k^2 – k\)
We get: \(k^2 – k = 110\)
Subtract \(110\) from both sides to get: \(k^2 – k - 110 = 0\)
Factor: \((k - 11)(k + 10) = 0\)
So, EITHER \(k = 11\) OR \(k = -10\)
Since \(k\) cannot be negative, it must be the case that \(k = 11\)
In other words, \(S_{11} = 110\)

The question asks us to find the value of \(S_{n – 1}\)?
Since \(n = 11, \), this is asking us to find the value of \(S_{11 – 1}\) which is the same as the value of \(S_{10}\)

Apply the formula again to get: \(S_{10} = 10^2 – 10 = 100 - 10 = 90\)

Answer: D

Cheers,
Brent

D.

Sk=k^2–k = K(K-1) i.e. Sum of two consecutive integers i.e. (k-1 ) & k

Sn = 110 = 11 * 10. i.e. n =11

n-1 = 10

Sn-1 = 10 * 9 = 90
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\(S_n = 110\)
\(110 = n^2 - n\)
\(110 = n(n-1)\)
\(n = 11\)
\(n - 1 = 10\)

\(10^2 - 10 = 90\)

Answer is D.
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Solution

Given
In this question, we are given that

• A sequence is defined by the general term \(S_k = k^2 – k\), for all positive integers k
• The value of \(S_n\) is 110

To find
We need to determine

• The value of \(S_{n-1}\)

Approach and Working out
As per the given definition,

• \(S_n = n^2 – n = 110\)
Or, \(n^2 – n – 110 = 0\)
Or, \(n^2 – 11n + 10n – 110 = 0\)
Or, (n – 11) (n + 10) = 0

Hence, n = 11 (as n cannot be negative)

Therefore, \(S_{n-1}\) = \(S_{10}\) = \(10^2 – 10\) = 100 – 10 = 90

Thus, option D is the correct answer.

Correct Answer: Option D
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