Standard +0.3 This is a straightforward application of standard GP and AP sum formulas with clear given values. Students must use S_n = a(r^n - 1)/(r - 1) for the GP, equate it to S_n = n/2(2a + (n-1)d) for the AP, then solve for the first term. The arithmetic is slightly involved but requires no insight beyond direct formula application, making it slightly easier than average.
6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.
\(S_6\) for GP \(= 192(1.5^6 - 1) \div 0.5 = 3990\)
M1
Correct sum formula used.
\(S_{21}\) for AP \(= \frac{21}{2}(2a + 20 \times 1.5)\)
M1 DM1 A1
Correct sum formula used. Needs both M's - soln of sim eqns. CAO
Equate and solve \(\Rightarrow a = 175\)
M1 A1
Correct formula used.
\(21^{\text{st}}\) term in AP \(= a + 20d = 205\) (or from \(3990 = 21(a + l)/2\))
[6]
$\text{GP: } a = 192, r = 1.5, n = 6$
$\text{AP: } a = a, d = 1.5, n = 21$
$S_6$ for GP $= 192(1.5^6 - 1) \div 0.5 = 3990$ | M1 | Correct sum formula used.
$S_{21}$ for AP $= \frac{21}{2}(2a + 20 \times 1.5)$ | M1 DM1 A1 | Correct sum formula used. Needs both M's - soln of sim eqns. CAO
Equate and solve $\Rightarrow a = 175$ | M1 A1 | Correct formula used.
$21^{\text{st}}$ term in AP $= a + 20d = 205$ (or from $3990 = 21(a + l)/2$) | [6] |
6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.
\hfill \mbox{\textit{CAIE P1 2005 Q6 [6]}}