eg \(192 = 2a + 29(\text{``}{6}\text{''})\) oe or \(123 = a + 19(\text{``}{6}\text{''})\) oeor eg \(192 = 2(\text{``}{9}\text{''}) + 29d\) oe or \(123 = \text{``}{9}\text{''} + 19d\) oe
M1
Working required Answer: \(a = 9\) \(d = 6\)
A1
(5)
(5 marks)
Notes
M1: for using \(U_n = a + (n - 1)d\)
M1: for using \(S_n = \dfrac{n}{2}\left(2a + (n - 1)d\right)\)
M1: (dep on M2) for a correct method to find \(a\) or \(d\):
coefficients of \(a\) or \(d\) the same in correct equations and correct operator to eliminate selected variable resulting in an equation in \(a\) only or in \(d\) only
or
writing \(a\) or \(d\) in terms of the other variable and correctly substituting (condone missing brackets)
M1: (dep on M3) for substituting their found value of \(a\) or \(d\) into a correct equation
A1: dep on M2 \(a\) and \(d\) must be clearly identified
M1: for a correct expression or equation using the common difference, may be in terms of \(d\) for this mark we will allow an expression for \(-d\) or \(-2d\)
M1: for a correct equation for the sum of 9 terms in \(x\) and \(d\) or in terms of \(x\) and \(y\) or in terms of \(x\) or in terms of \(d\)
for “\(3y - 2x - 9\)” we will allow \((3y - 4) - (2x + 5)\) or for using their incorrect simplification from \((3y - 4) - (2x + 5)\) shown
for “\(4x - 3y + 2\)” we will allow \((4x - 2) - (3y - 4)\) or for using their incorrect simplification from \((4x - 2) - (3y - 4)\) shown
similarly for their “\(x - 3.5\)” and their “\(d + 3.5\)”
M2:left hand column 2 correct equations in terms of \(x\) and \(y\) in the form \(px + qy = r\) oe or 2 correct equations in terms of \(x\) and \(d\) in the form \(px + qd = r\) oe
right hand column 2 correct equations in terms of \(x\) and \(y\) where one is substituted into the other to get a correct equation in the form \(px + q = r\) or \(py + q = r\) or 2 correct equations in terms of \(x\) and \(d\) where one is substituted into the other to get a correct equation in the form \(px + q = r\) or \(pd + q = r\)
If not M2 then M1 for one correct equation in any of the required forms from the left hand or right hand column
A2: (dep on M2) oe (allow 7.3(33…))
(A1 for \(x = \dfrac{11}{2}\) or \(y = \dfrac{22}{3}\))
