Moderate -0.5 This is a straightforward two-equation system using standard arithmetic sequence formulas (a + 3d = 3 and S₆ = 27). It requires routine algebraic manipulation with no conceptual challenges, making it slightly easier than average but not trivial since it involves simultaneous equations and the sum formula.
4. The \(4 ^ { \text {th } }\) term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27
Find the first term and the common difference of this sequence.
M1A1: Writes correct (possibly un-simplified) equation for 4th term or sum of first 6 terms
\(a+3d=3\) AND \(\frac{6}{2}(2a+5d)=27\)
A1
A1cao: Correct equation for 4th term and correct equation for sum
Eliminates one variable to find \(a\) or \(d\) from 2 equations
dM1
Eliminates one variable from two equations in \(a\) and \(d\); implies previous M mark
\(a=12\) or \(d=-3\)
A1
One variable correct
\(a=12\) and \(d=-3\)
A1
Both variables correct
Note: If both equations correct and no working and values of \(a\) and \(d\) both incorrect, scores dM0. If equations incorrect and values obtained with no working, also scores dM0.
## Question 4:
| Working | Mark | Guidance |
|---------|------|----------|
| $a+3d=3$ **OR** $\frac{6}{2}(2a+5d)=27$ | M1 A1 | M1A1: Writes correct (possibly un-simplified) equation for 4th term **or** sum of first 6 terms |
| $a+3d=3$ **AND** $\frac{6}{2}(2a+5d)=27$ | A1 | A1cao: Correct equation for 4th term **and** correct equation for sum |
| Eliminates one variable to find $a$ or $d$ from 2 equations | dM1 | Eliminates one variable from two equations in $a$ and $d$; implies previous M mark |
| $a=12$ **or** $d=-3$ | A1 | One variable correct |
| $a=12$ **and** $d=-3$ | A1 | Both variables correct |
**Note:** If both equations correct and **no working** and values of $a$ and $d$ both incorrect, scores dM0. If equations incorrect and values obtained with no working, also scores dM0.
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4. The $4 ^ { \text {th } }$ term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27
Find the first term and the common difference of this sequence.\\
\hfill \mbox{\textit{Edexcel C12 2016 Q4 [6]}}