In the following question, a number series is given in which one term is missing. Find out the missing term. 15, 63, 143, 255, ?, 575, 783

Ο 301

Ο 401

399

Ο 299

To identify the missing term in the series \( 15, 63, 143, 255, ?, 575, 783 \), let's analyze the pattern between the terms.

First, let's calculate the differences between successive terms:

\[
63 - 15 = 48
\]
\[
143 - 63 = 80
\]
\[
255 - 143 = 112
\]
\[
575 - 255 = 320
\]
\[
783 - 575 = 208
\]

The differences are \( 48, 80, 112, ?, 320, 208 \).

Next, we analyze the pattern in the differences:

\[
80 - 48 = 32
\]
\[
112 - 80 = 32
\]
\[
320 - 112 = 208 \quad \text{(This does not fit the previous pattern)}
\]

However, the differences between the terms do not follow a simple arithmetic progression. Therefore, let's reconsider the approach. Another potential pattern could be related to a polynomial sequence.

Considering the terms are closely spaced but not consistent, an alternate approach might be required. Let's derive if these numbers follow a polynomial pattern or any specific pattern:

Another way is to look for a hidden polynomial sequence. However, checking for a cubic polynomial:

Given \( a_n = an^3 + bn^2 + cn + d \):

1st term: \( 15 = a(1)^3 + b(1)^2 + c(1) + d \)
2nd term: \( 63 = a(2)^3 + b(2)^2 + c(2) + d \)
3rd term: \( 143 = a(3)^3 + b(3)^2 + c(3) + d \)
4th term: \( 255 = a(4)^3 + b(4)^2 + c(4) + d \)

Let's solve these equations:

For 1:
\[
a + b + c + d = 15
\]

For 2:
\[
8a + 4b + 2c + d = 63
\]

For 3:
\[
27a + 9b + 3c + d = 143
\]

For 4:
\[
64a + 16b + 4c + d = 255
\]

Solve these equations using matrix method or systematic elimination for \( a, b, c, \) and \( d \):

Subtract equations to eliminate d:

\[
7a + 3b + c = 48
\]
\[
19a + 5b + c = 80
\]
\[
37a + 7b + c = 112
\]

Subtract these to find constant difference eliminating c:

\[
12a + 2b = 32 \implies 6a + b = 16 \implies b = 16 - 6a
\]

Substitute \( b = 16 - 6a \):

\[
19a + 5(16 - 6a) + c = 80 \implies 19a + 80 - 30a + c = 80 \implies -11a + c = 0 \implies c = 11a
\]

\[
7a + 3(16 - 6a) + 11a = 48 \implies 7a + 48 - 18a + 11a = 48 \implies 0 = 48\]

Verify sequence, find constant pattern check:

Using options substitution check \( 299, 301, 399 \):

\[ 399 - 255 = 144 575 - 399 = 176\]

Check fits original patterns terms:

\[ 255, 399, 575 has consistent growth higher differences close to polynomial cubic fitting. Thus verifying with difference fits polynomial model close consistent gaps increase. Answer: **399**

Discover how to find the missing term in the number series 15, 63, 143, 255, ?, 575, 783 using pattern recognition and polynomial sequence analysis. Suitable for advanced high school students and above.

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