Sequences & Series · Ordinary Level

Finding a and d from Two Terms

Ordinary Level · Class 5 · Tap NEXT to begin

Section 1 of 1

Two Terms, Two Unknowns

Make two equations

Given two terms, write each with $T_n = a + (n-1)d$. That gives two equations — subtract to find $d$, then back-substitute for $a$.

In an arithmetic sequence $T_2 = 5$ and $T_6 = 13$. Find $a$ and $d$.

$T_2 = a + d = 5$; $T_6 = a + 5d = 13$

Subtract: $-4d = -8 \Rightarrow d = 2$

$a + 2 = 5 \Rightarrow a = 3$

$a = 3,\ d = 2$

$T_4 = 19$ and $T_7 = 31$. Find $a$ and $d$.

$a + 3d = 19$; $a + 6d = 31$

Subtract: $3d = 12 \Rightarrow d = 4$

$a + 12 = 19 \Rightarrow a = 7$

$a = 7,\ d = 4$

You try

$T_5 = 13$ and $T_8 = 22$. Find $a$ and $d$.

$a + 4d = 13$ and $a + 7d = 22$.

Subtract: $3d = 9 \Rightarrow d = 3$

$a + 12 = 13 \Rightarrow a = 1$

$a = 1,\ d = 3$

You try

$T_3 = 15$ and $T_{12} = 39$. Find $a$ and $d$.

$a + 2d = 15$ and $a + 11d = 39$.

Subtract: $9d = 24 \Rightarrow d = \tfrac{8}{3}$

$a + 2(\tfrac{8}{3}) = 15 \Rightarrow a = 15 - \tfrac{16}{3} = \tfrac{29}{3}$

$a = \tfrac{29}{3},\ d = \tfrac{8}{3}$

Two terms → two equations → $a$ and $d$. Class 6: adding the terms up — series.