A half-tail identity and triangular rows
(a) The first 2 terms of a GP are , where . If the sum of the first terms equals the sum to infinity of the remaining terms, prove . marks
(b) The AP is arranged in rows: 1st row: 2; 2nd: 4,6; 3rd: 8,10,12; 4th: 14,16,18,20; … (native table below).
(i) Show the first term in row is . marks
(ii) Find the sum of all terms from row 1 to row . marks
[2009/DHS/Prelim/I/9]
Solution
Equal partial sum and tail
Use the equality of the first terms and the remaining infinite tail to prove .
Hint
Use , then cancel the common prefactor.
Answer · (a)
Equality of the partial sum and tail gives , hence .
Nonzero convergent GP.
Express the equality
Equal halves of the total mean twice the partial sum equals the full sum.
Cancel permitted factors
The given p≠0 and convergence make the denominators nonzero.
Clear the power of p
This produces the requested relationship without assuming p positive.
First term in row n
Show that row begins with .
Hint
Count entries before the row; then add one for its first position.
Answer · (b)(i)
.
Count previous entries
The nth row begins immediately after these entries.
Convert position to value
The original AP is 2,4,6,... so position j has value 2j.
Alternative: sum the jumps between row starts
The row starts are , whose jumps are . Sum the first jumps and add the initial value 2. For there are no jumps, and the first entry is 2.
Sum rows before n
Find the sum of all entries in rows 1 through .
Hint
The last entry before row n is not the first entry of row n.
Answer · (b)(ii)
.
Find the last included value
The next row starts at n²−n+2; the previous value is two less.
Use first-plus-last
There are N terms with first 2.
Translate row positions into sequence indices
Although arranged as a table, all entries before row n remain the first N terms of 2,4,6,... .