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the sum of a sequence of consecutive integers is 342. the largest integer in the sequence is 3 times greater than the smallest integer in the sequence. what is the smallest integer and how many integers are in the sequence?

smallest integer=9,

total number of integers are in the sequence  = 19.

Step-by-step explanation:

[ where, l: last term]

[where, d: common difference]

using the first formula,

342 =

342=

342=2an  .......equation 1.

using the second formula,

342=

684=n[2a+n-1]

684=2an+n^2-n

from equation 1,

342=n^2-n   .......equation 1

-n-342=0

after solving, the solution to the above equation is 19.

substituting this in equation 2,

171=a[19]

a=9

so, smallest integer= a = 9,

total number of integers are in the sequence = n = 19.

Smallest integer is 9

number of integers is 19

Step-by-step explanation:

Step-by-step explanation:

Sum on consecutive integers is

n(a+l)/2

where a is the first/smallest integer, l is the last/largest integer and n is the number of integers

n(a+l)/2 = 342 (1)

l = 3a (2)

l = a + (n-1) (3)

Using (2) and (3)

3a = a+(n-1)

2a = n-1

n = 2a+1 (4)

Using (1) (2) and (4)

(2a+1)(a+3a)/2 = 342

(2a+1)(4a) = 684

4(2a²+a) = 684

2a²+a= 684÷4

2a²+a = 171

2a²+a-171 = 0

2a²+19a-18a-171 = 0

a(2a+19)-9(2a+19) = 0

(2a+19)(a-9) = 0

a = 9 or a = -19/2

Since a is an integer, it has to be 9

a = 9

n = 2a+1 = 2(9)+1

n = 19

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

yes

step-by-step explanation:

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