High School

If the sum of n terms of an arithmetic progression (AP) is given by [tex]S_n = abn + cn^2[/tex], where [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are constants independent of [tex]n[/tex], then:

A. The common difference of the AP is [tex]a[/tex].
B. The first term of the AP is [tex]b[/tex].
C. The common difference of the AP is [tex]c[/tex].
D. The sum of the first [tex]n[/tex] terms of the AP is [tex]abn + cn^2[/tex].

Answer :

Final answer:

The common difference of the Arithmetic Progression (AP) in the given scenario is c.

Explanation:

The correct statement is: c) The common difference of the AP is c

To explain this, we note that in an Arithmetic Progression (AP), the formula for the sum of the first n terms is Sₙ = ½n[2a + (n-1)c], where a is the first term and c is the common difference. Comparing this with Sₙ = abncn², we see that c = 1, hence the common difference is indeed c.

The common difference is not directly given by the terms a, b, or c. Instead, the common difference can be deduced by comparing the expressions for Sn and identifying the term that corresponds to d. Hence, given the sum in the form Sn = abncn2, none of the options a) a, b) b, or c) c individually corresponds to the common difference of the AP without further information.

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