Find the incorrect number in the series: 18, 20, 23, 32, 48, 73, 109.

Question

ScanSolve AI Tutor · Accepted Answer

To find the incorrect number in the series, we will analyze the differences between consecutive terms.

The given series is: $18, 20, 23, 32, 48, 73, 109$.

Calculate the differences between consecutive terms.
$20 - 18 = 2$ $23 - 20 = 3$ $32 - 23 = 9$ $48 - 32 = 16$ $73 - 48 = 25$ $109 - 73 = 36$
The sequence of differences is $2, 3, 9, 16, 25, 36$ .
Identify the pattern in the differences.
We observe that the last four differences are perfect squares:
$9 = 3^2$ $16 = 4^2$ $25 = 5^2$ $36 = 6^2$
This suggests that the pattern of differences should be consecutive squares starting from $1^2$ .
Determine the expected sequence of differences.
The expected differences are $1^2, 2^2, 3^2, 4^2, 5^2, 6^2$ , which are $1, 4, 9, 16, 25, 36$ .
Reconstruct the series using the first term and the expected differences to find the incorrect number.
- The first term is 18.
- The second term should be $18 + 1^2 = 18 + 1 = 19$ . (The given second term is 20).
- The third term should be $19 + 2^2 = 19 + 4 = 23$ . (The given third term is 23, which matches).
- The fourth term should be $23 + 3^2 = 23 + 9 = 32$ . (The given fourth term is 32, which matches).
- The fifth term should be $32 + 4^2 = 32 + 16 = 48$ . (The given fifth term is 48, which matches).
- The sixth term should be $48 + 5^2 = 48 + 25 = 73$ . (The given sixth term is 73, which matches).
- The seventh term should be $73 + 6^2 = 73 + 36 = 109$ . (The given seventh term is 109, which matches).
Comparing the given series ( $18, 20, 23, 32, 48, 73, 109$ ) with the reconstructed series ( $18, 19, 23, 32, 48, 73, 109$ ), the number 20 is incorrect as it should be 19 to follow the pattern.

Answer