Use the formula for the sum of the first n integers either to evaluate the sum or to express it in closed form. a. 9 + 10 + 11 + 12 + + 400
b. 9 + 10 + 11 + 12 + + k

A) 80,164

B) S_n = (n/2)(17 + n)

Step-by-step explanation:

A) Formula for the nth term of an arithmetic progression series is;

a_n = a + (n - 1)d

a is first term of the series and in this case it is 9.

Difference is d which is 1 since they are consecutive integers.

Thus, For the last term which is 400, we have;

400 = 9 + (n - 1)1

400 - 9 = n - 1

391 = n - 1

n = 391 + 1

n = 392

Thus,there 392 terms in this sequence.

Formula for sum of an AP is;

S_n = (n/2)(2a + (n - 1)d)

S_392 = (392/2)((2 × 9) + (392 - 1)1)

S_392 = 80,164

B) The last term here is k, so we can't know exactly how many terms are in this sequence. Thus, let the number of terms be n.

Now, as above,

a = 9

d = 1

Sum of this AP sequence is;

S_n = (n/2)((2 × 9) + (n - 1)1)

S_n = (n/2)(18 + n - 1)

S_n = (n/2)(17 + n)

answer: sine

step-by-step explanation:

let m = monomial

let t = trinomial

so, m times t = t.

monomial times a trinomial = another trinomial.