Question 1152187
A quadratic pattern has a second term equal to -1,a third equal to -6 and a fifth term equal to -14.Calculate the second difference of this quadratic pattern.
Hence,or otherwise ,calculate the first term of the pattern.
<pre>This person makes these problems so, so, COMPLEX and more time-consuming than is necessary.
After substituting each term into the quadratic pattern formula: {{{matrix(1,3, a[n], "=", An^2 + Bn + C)}}}, you should get the following system of equations:
 4A + 2B + C = - 1 ------ eq (i)
 9A + 3B + C = - 6 -------eq (ii)
25A + 5B + C = - 14 ----- eq (iii)

5A + B = - 5 ------- Subtracting eq (i) from eq (ii) ----- eq (iv)
16A + 2B = - 8 ----- Subtracting eq (ii) from eq (iii) --- eq (v)
- 10A - 2B = 10 ---- Multiplying eq (iv) by  -  2 ----- eq (vi)
6A = 2 ----- Adding eqs (v) & (vi)
{{{matrix(1,5, A, "=", 2/6, "=", 1/3)}}}

{{{matrix(1,3, 5(1/3) + B, "=", - 5)}}} ----- Substituting {{{1/3}}} for A in eq (iv)
{{{matrix(1,3, 5/3 + B, "=", - 5)}}}
{{{matrix(1,7, B, "=", - 5  -  5/3, "=", - 15/3  -  5/3, "=", - 20/3)}}}

{{{matrix(1,3, 4(1/3) + 2(- 20/3) + C, "=", - 1)}}} --- Substituting {{{1/3}}} for A, and {{{- 20/3}}} for B in eq (i)
{{{matrix(1,3, 4/3  -  40/3 + C, "=", - 1)}}}
{{{matrix(1,3, - 36/3 + C, "=", - 1)}}}
- 12 + C = - 1 
C = - 1 + 12
C = 11

{{{highlight_green(matrix(1,7, Second, "difference:", 2A, "=", 2(1/3), "=", 2/3))}}}    {{{highlight_green(matrix(1,9, First, "term:", A + B + C, "=", 1/3 - 20/3 + 11, "=", 1/3 - 20/3 + 33/3, "=", 14/3))}}}
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Although not needed, if you're curious, the quadratic sequence, {{{highlight_green(matrix(1,3, a[n], "=", An^2 + Bn + C))}}} becomes: {{{highlight_green(matrix(1,3, a[n], "=", (1/3)n^2  -  (20/3)n + 11))}}}</pre>