Question

Please help lots of points rewarded

Answer 1

Answer:

• tₙ =  2n² - 3

Step-by-step explanation:

Standard form of a quadratic equation:

• y = ax² + bx + c

Apply same for the quadratic sequence:

• tₙ = an² + bn + c

Substitute n with 1, 2 and 3 to get the system:

• -1 = a*1² + b*1 + c  ⇒ a + b + c = -1
• 5 = a*2² + b*2 + c  ⇒ 4a + 2b + c = 5
• 5 = a*3² + b*3 + c  ⇒ 9a + 3b + c = 15

Subtract the first equation from the second and subtract the second equation from the third to get:

• 3a + b = 6
• 5a + b = 10

Again subtract the equations to find a:

• 2a = 4
• a = 2

Then find b and c:

• b = 3a - 6 = 6 - 6 = 0
• c = -1 - a - b = -1 - 2 - 0 = -3

Substitute the coefficients back into the equation of the quadratic sequence:

• tₙ = 2n² + 0n + (-3) = 2n² - 3

Answer 2

A of n = 2n^2 - 3
the nth term = 2 times n to the power of 2 minus 3