With Ends

With Ends

What is it?

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

In simple words

• If last term l is given, S_n = n(a+l)/2.
• 1 to n: n(n+1)/2
• Find n from last term
• State the rule, show the working, and check the answer.

Important terms

WITH ENDS: If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

WITH ENDS: If last term l is given, S_n = n(a+l)/2.

What you should learn

• State the rule or meaning of with ends in easy words.
• Apply with ends in a clear step-by-step solution.
• Check the conditions, working, and final answer for accuracy.

Examples

• 1 to n: n(n+1)/2
• Find n from last term
• Check a_n

MAIN IDEA

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

EXAM USE

Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

Full school-level explanation

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

Understand the idea, then apply it:

WHAT — WITH ENDS

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

WHY — Why it matters

This rule is a separate learning unit inside sum of an arithmetic series. It can be defined, applied, and checked in its own FBISE question.

HOW — How to answer

Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

MAIN IDEA

If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats in a theatre, logs in a stack (school).

Remember

• average of ends * n

Examples

• 1 to n: n(n+1)/2
• Find n from last term
• Check a_n

EXAM USE

Write the rule first: If last term l is given, S_n = n(a+l)/2. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.

Remember

• focused answer
• accurate support

Examples

• State the central idea
• Use one accurate detail
• Check the final response

How to answer a with ends question

How it works

• Read the exact command word and identify what the question asks.
• State the matching idea, rule, or meaning before adding detail.
• Use the unit evidence or mathematical working that directly supports the answer.
• Check accuracy, relevance, and answer length before finishing.

Remember

• average of ends * n
• S_n = n/2 * (2a+(n-1)d)
• also n/2 * (a1+a_n)
• do not drop /2
• average of ends * n
• focused answer
• accurate support
• average of ends * n

Deeper understanding

Go beyond memorising With Ends. Connect the definition to reasons, relationships, exceptions, and exam wording.

Common mistakes

Exam focus

Core Idea

What to Know

Example or Evidence

How to Answer

Exam focus

Quick revision

• average of ends * n
• S_n = n/2 * (2a+(n-1)d)
• also n/2 * (a1+a_n)
• do not drop /2
• WITH ENDS: If last term l is given, Sn = n(a+l)/2. Find n first from an = l if needed. Word problems: seats i
• WITH ENDS: If last term l is given, S_n = n(a+l)/2.

S_n uses n/2, not n. Forgetting /2 doubles the answer.

For long answers on With Ends: define clearly → explain the main parts → give one example → state one exam caution.