A quadratic sequence has the following properties: - The second difference is 10 - NSC Mathematics - Question 3 - 2023 - Paper 1

To show the general term of the quadratic sequence, we first denote the terms as:

• First difference: $T_n = an^2 + bn + c$
• Second difference: $2a = 10$, thus $a = 5$.

Next, using the condition $T_1 = T_2$:

$T_1 = 5(1)^2 + b(1) + c$ $T_2 = 5(2)^2 + b(2) + c$

Equating gives:

$5 + b + c = 20 + 2b + c$

This simplifies to:

$b = -15$

Now we use the third property:

$T_1 + T_2 + T_3 = 28$

Calculating each term gives:

• $T_1 = 5(1)^2 - 15(1) + 16 = 6$
• $T_2 = 5(2)^2 - 15(2) + 16 = 6$
• $T_3 = 5(3)^2 - 15(3) + 16 = 10$

Summing these gives:

$6 + 6 + 10 = 28$

So the expression for $T_n$ becomes:

$T_n = 5n^2 - 15n + 16$