State the value of the discriminate. Then determine the number of real roots of the equation.

n(7n+8)=-10

no real solutions

Step-by-step explanation:

Given a quadratic equation in standard form

ax² + bx + c = 0 : a ≠ 0

Then the nature of the roots can be determined from the discriminant

b² - 4ac

• If b² - 4ac > 0 then 2 real and distinct roots

• If b² - 4ac = 0 then 2 real and equal roots

• If b² - 4ac < 0 then no real roots

Given

n(7n + 8) = - 10 ← distribute left side

7n² + 8n = - 10 ( add 10 to both sides )

7n² + 8n + 10 = 0 ← in standard form

with a = 7, b = 8 and c = 10, thus

b² - 4ac = 8² - (4 × 7 × 10) = 64 - 280 = - 216

Since b² - 4ac < 0 the equation has no real roots

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for

step-by-step explanation: