A-level · Beta

Sequences and series

Change terms and compare arithmetic and geometric partial sums.

Drag A to change the first term and D to change the next term. Blue bars are terms; the green line shows partial sums.

u_n=a+(n-1)d,\quad S_n=\frac n2[2a+(n-1)d]

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Equal differences change the blue terms by a fixed amount.

Green points are finite partial sums, not individual terms. First term u₁ = a. Displayed decimals are rounded.

Predict, test and explain

Step 1 of 3

Predict

What is the sum of the first four terms when a = 3 and r = ½?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

What is the sum of the first four terms when a = 3 and r = ½?

$S_4=3+\frac32+\frac34+\frac38$

Hint

Start with u₁ = a; the nth exponent is n − 1.

Reveal answer

The terms are 3, 3/2, 3/4 and 3/8. Their finite sum is 45/8, below the infinite sum 6.

$\frac{45}{8}$

Hint 1

Start with u₁ = a; the nth exponent is n − 1.

Hint 2

A sequence term and a series sum are different quantities.

Hint 3

The nonzero infinite geometric series converges only for |r| < 1.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the sum of 3 + 3/2 + 3/4 + 3/8.

Answer: Finite sum 45/8; infinite sum 6.

Connect this idea

Watch out: The infinite sum is not the finite sum of four terms.

Specification and learning route

AQA D3, D4, D5 · Edexcel Pure 4.3, Pure 4.4, Pure 4.5

A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Nth-term notation; Indices; Algebra; Sigma notation.