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Mathematics, 07.07.2020 01:01 alexabessin

If the sum of the integers from 15 to 50, inclusive, is equal to the sum of the integers
from n to 50, inclusive, and n<15, then n=

A)-49
B)-35
C)-15
D-14

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Answers: 1

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Answers

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Answer from: EMQPWE

n = -15

Step-by-step explanation:

Given

Sum of Integers from 15 to 50 (inclusive) = Sum of integers from n to 50 (inclusive)

n < 15

Required

Find n

We can split the given parameters to 2

i. Sum of Integers from 15 to 50 (inclusive)

ii. Sum of integers from n to 50 (inclusive)

Solving for (i): Sum of Integers from 15 to 50 (inclusive)

We'll make use of sum of n terms of an arithmetic;

This is given as follows

$S_n = \frac{n}{2}(T_1 + T_n)$

Where n is the number of terms from 15 to 50

T1 is the first term; T1 = 15

Tn is the last term; Tn = 50

n is calculated using

n = T_n - T_1 + 1

n = 50 - 15 + 1

n = 36

The formula becomes

$S_n = \frac{n}{2}(T_1 + T_n)$

$S_n = \frac{36}{2}(15 + 50)$

$S_n = \frac{36}{2}(65)$

S_n = 18(65)

S_n = 1170

Solving for (ii): Sum of integers from n to 50 (inclusive)

We'll also make use of the same formula used above

$S_n = \frac{n}{2}(T_1 + T_n)$

Where n is the number of terms from n to 50

T1 is the first term; T1 = n

Tn is the last term; Tn = 50

n is calculated using

n = T_n - T_1 + 1

n = 50 - n + 1

n = 50 + 1 - n

n = 51 - n

The formula becomes

$S_n = \frac{n}{2}(T_1 + T_n)$

$S_n = \frac{51 - n}{2}(n + 50)$

Recall that the Sum of Integers from 15 to 50 (inclusive) = Sum of integers from n to 50 (inclusive);

This implies that

Sn (i) = Sn (ii)

As such; we have

$S_n = \frac{51 - n}{2}(n + 50) = 1170$

$\frac{51 - n}{2}(n + 50) = 1170$

Multiply both sides by 2

$2 * \frac{51 - n}{2}(n + 50) = 1170 * 2$

(51 - n)(n + 50) = 1170 * 2

(51 - n)(n + 50) = 2340

Open Brackets

51(n + 50) - n(n + 50) = 2340

51*n + 51 * 50 - n * n - n * 50 = 2340

51n + 2550 - n^2 - 50n = 2340

Reorder

-n^2 + 51n - 50n + 2550 = 2340

-n^2 + n + 2550 = 2340

Subtract 2340 from both sides

-n^2 + n + 2550 -2340 = 2340 -2340

-n^2 + n + 210 = 0

Multiply both sides by -1

-1(-n^2 + n + 210) = -1 * 0

n^2 - n - 210 = 0

At this point, we have a quadratic equation; as such, it'd be solved as follows:

n^2 - n - 210 = 0

Expand

n^2 + 15n - 14n - 210 = 0

Factorize

n(n + 15) - 14(n + 15) = 0

(n - 14)(n + 15) = 0

Split the above expression

$(n - 14) = 0 \ or\ (n + 15) = 0$

$n - 14 = 0 \ or\ n + 15 = 0$

$n = 14\ or\ n = -15$

The question states that n < 15;

This means that we'll discard the value of n = 14

Hence, n = -15

ansver

Answer from: Quest

3432532

step-by-step explanation:

ansver

Answer from: Quest

the system of inequalities is

$y> (1/3)x$

$y\leq-x+4$

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

$m=y/x=1/3$

the equation of the dashed line is

$y=(1/3)x$

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

$y> (1/3)x$

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

$m=(4-0)/(0-4)=-1$

the equation of the solid line is

$y=-x+4$

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

$y\leq-x+4$

step 3

the system of inequalities is

$y> (1/3)x$

$y\leq-x+4$

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