Let $\left\{x_n\right\}$ be a sequence of positive real numbers such that $x_n = \sqrt{x_{n-1}x_{n-2}}$ for all $n\geq3$. Then show that the sequence converges to $(x_1x_2^2)^{1/3}$.   How can I do the sum? I have learnt Squeeze theorem, ratio test and root test for convergence of sequence, but I am not sure whether those theorems will help in solving this sum. Please give some hint to approach it.  Please anyone help me solve it. Thanks in advance.
We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit.
Automatic vacuum fails with `ERROR: XX000: cache lookup failed for type 0` I've been noticing something like this popping up every 15 seconds in my server logs:  2020-01-30 21:10:30 UTC::@:[24969]:ERROR: XX000: cache lookup failed for type 0 2020-01-30 21:10:30 UTC::@:[24969]:CONTEXT: automatic vacuum of table "myschema.mytable" 2020-01-30 21:10:30 UTC::@:[24969]:LOCATION: get_typlenbyval, lsyscache.c:2036   When I manually run a VACUUM ANALYZE myschema.mytable; it runs without any error, and the error in my logs goes away. However, inevitably, it returns again at some point.   The table in question is defined as follows:  tfs_dev=&gt; \d myschema.mytable;                                                    Table "myschema.mytable"    Column    |   Type    | Collation | Nullable |                                  Default -------------+-----------+-----------+----------+---------------------------------------------------------------------------  level       | text      |           | not null |  id          | integer   |           | not null | nextval('myschema.mytable_id_seq'::regclass)  text_id     | text      |           |          |  name        | text      |           |          |  geom        | geometry  |           |          |  valid_dates | tstzrange |           |          | tstzrange(NULL::timestamp with time zone, NULL::timestamp with time zone)  adjacent    | integer[] |           |          |  valence     | integer   |           |          |  color       | smallint  |           |          | 0 Indexes:     "mytable_pkey" PRIMARY KEY, btree (level, id)     "mytable_text_id_key" UNIQUE CONSTRAINT, btree (text_id)     "mytable_gix" gist (level, geom, id, text_id, name)     "mytable_spgix" spgist (geom)     "mytable_text_id_ix" btree (text_id, level, id, name) Referenced by:     TABLE "otherschema.othertable" CONSTRAINT "othertable_mytable_fk" FOREIGN KEY (level, level_id) REFERENCES myschema.mytable(level, id)     TABLE "otherschema.othertable2" CONSTRAINT "othertable2_mytable_fk" FOREIGN KEY (level, level_id) REFERENCES myschema.mytable(level, id)   Many other posts with a similar error seem to be about a foreign data wrapper, which is present in this database, but not with this table/schema. 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