PYQ Vault
Ebooks

SPI Mathematics · Sequence and Series

Three Terms in an AP

If a, b, c are in AP, the middle term is the average of the other two (2b = a + c), and three unknown terms are best written as a − d, a, a + d.

All Sequence and Series notes

Why this matters

Five questions in the bank, 4 EASY and 1 MODERATE. Each turns on one of two facts: twice the middle term equals the sum of the other two, and the sum of three terms centred on the middle one is three times the middle term.

Concept 1 of 2: The middle term is the average of its neighbours

If a, b, c are in AP, the step from a to b equals the step from b to c. So b sits exactly halfway between them, and twice b is a plus c. Use this whenever three terms are written with an unknown in them.

Definition

  • a,b,ca, b, c are in AP exactly when b−a=c−bb - a = c - b, that is 2b=a+c2b = a + c.
  • b=a+c2b = \dfrac{a + c}{2} is called the arithmetic mean of aa and cc.
  • With an unknown in the terms, write 2×middle=first+third2 \times \text{middle} = \text{first} + \text{third} and solve. Then check that the gaps are equal.

Three terms in AP

2b=a+cb=a+c22b = a + c \qquad b = \frac{a + c}{2}

Worked example

Find xx if x+1x + 1, 2x+12x + 1 and 4x−34x - 3 are in AP.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Sequence and Series · Three Terms in an AP

If the first three terms of an A.P. are 3p - 1, 3p + 5, 5p + 1 respectively; then the value of p is:

Writing the middle term as the sum

The middle term is the average of its neighbours, not their sum. For 4, 10, 16: 2×10=4+162 \times 10 = 4 + 16, while 4+16=204 + 16 = 20 is not 10.

Skipping the check

After solving for the unknown, put it back into all three terms. If the gaps are not equal, an algebra slip has crept in.
Drill 1 more on the middle term is the average of its neighbours

Concept 2 of 2: Three unknown terms written around the middle one

When a question gives the sum of three AP terms, call them a minus d, a and a plus d. The two d's cancel in the sum, so the sum alone gives the middle term.

Definition

  • Write three terms in AP as a−d,a,a+da - d, a, a + d. Their sum is 3a3a.
  • So the middle term is the sum divided by 3, whatever dd is.
  • A product condition then gives dd, using (a−d)(a+d)=a2−d2(a - d)(a + d) = a^2 - d^2.
  • Choose the sign of dd from the wording: an increasing AP has d>0d > 0, a decreasing one d<0d < 0.

Three terms centred on the middle one

(a−d)+a+(a+d)=3a(a−d)(a+d)=a2−d2(a - d) + a + (a + d) = 3a \qquad (a - d)(a + d) = a^2 - d^2

Worked example

Three terms of an increasing AP add up to 27, and the product of the first and the third is 77. Find the three terms.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Sequence and Series · Three Terms in an AP

If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is

Calling the terms a, a + d, a + 2d

That also works, but the sum becomes 3a+3d3a + 3d and leaves two unknowns. Centre the terms on the middle one and the sum gives it at once.

Choosing the wrong sign for d

d2=9d^2 = 9 gives d=3d = 3 or d=−3d = -3. The two signs list the same numbers in opposite orders, so the first and third terms swap. Read 'increasing' or 'decreasing' before you answer.
Drill 2 more on three unknown terms written around the middle one

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Watch out for (4)

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paperDrill all 5 questions